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INDUSTRIAL SCIENCE DRAWING, 

ELEMENTARY 

PROJECTION DRAWING. 

THEORY AND PRACTICE. 



FOR PREPARATORY AND HIGHER SCIENTIFIC SCHOOLS; INDUSTRIAL AND NORMAL CLASSES; AND 
THE SELF-INSTRUCTION OF TEACHERS, INVENTORS, DRAFTSMEN, AND ARTISANS. 



In %i% ftJitrisions. 

DIV. I. ELEMENTARY PROJECTIONS. 

DIV. II. DETAILS OF MASONRY, WOOD, AND METAL CONSTRUCTIONS, 

DIV. ILL ELEMENTARY SHADOWS AND SHADING. 

DLV IV. ISOMETRICAL AND OBLIQUE PROJECTIONS, 

DIV. V. ELEMENTS OF MACHINES. 

DLV. VI. SIMPLE STRUCTURES AND MACHINES. 



By S. EDWAKD WAKKEN, C.E., 

i) 

FORMER PROFESSOR IN THE RENSSELAER POLYTECHNIC INSTITUTE, MASS. INST. OF TECHNOLOGY, 

AND BOSTON NORMAL ART SCHOOL; AND AUTHOR OF A SERIES OF TEXT-BOOKS 

IN DESCRIPTIVE GEOMBTRY AND INSTRUMENTAL DRAWING. 



Fifth Edition, Revised, and with a New Division on the Elements of Machines. 



i; iwmuiu, 



NEW YORK : 

JOHN WILEY & SONS, 

15 ASTOR PLACE. 

1880. 







Copyright, 1880, 
JOHN WILEY & SONS. 





PRESS OF 


s. 


W. GREEN'S SON, 




74 Beekman St., 




New York. 






CONTENTS 



Note to the Fourth Edition, ...... vi 

Note to the Fifth Edition, vii 

From the Original Preface, ix 

Preliminary Notes — Drawing Instruments and Materials, . xi 

DIVISION I.— PROJECTIONS. 

Chapter I. — First Principles. 

§ I. — The purely geometrical or rational theory of pro- 
jections, 1 

§ II. — Of the relations of lines to their projections, . 3 

§111. — Physical theory of projections, .... 5 

§ IV. — Conventional mode of representing the two planes of 
projection, and the two projections of any object 
- upon one plane; viz., the plane of the paper, . 5 

§ V. — Of the conventional direction of the light , and of the 

position and use of heavy lines, . 6 

§ VI.— Notation, 7 

§ VII. — Of the use of the method of projections, . . 8 

Chapter II. — Projection of Lines. Problems in Bight Projection; 
and including Projections showing two Sides of a 
Solid Might Angle. (Thirty-two Problems.) 
§ I. — Projections of straight lines, .... 9 

§ II. — Right projections of solids, 12 

§ III. — Projections showing two sides of a solid right angle 14 

§ IV. — Special Elementary Intersections and Developments 23 

General Examples, . . . . . . 31 

DIVISION II.— DETAILS OF MASONRY, WOOD, AND 
METAL CONSTRUCTIONS. 

Chapter I. — Constructions in Masonry. 

§ I. — General definitions and principles applicable both to 

brick and stone work, 33 

§ II.— Brick work, 33 

§ in.— Stone work, 36 

A stone box-culvert, ...... 37 



IV CONTEXTS. 

PAGE 

Chapter II. — Constructions in Wood. 

§ I. — General remarks. (Explanation of Scales.) . 39 

§ II. — Pairs of timbers whose axes make angles of 0° with 

each other, 42 

§ III. — Combinations of timbers whose axes make angles of 

90° with each other, 45 

§ IV. — Miscellaneous combinations. (Do welling, &c.) . 48 
§ V. — Pairs of timbers which are framed together obliquely 

to each other, 49 

§ VI. — Combinations of timbers whose axes make angles of 

180° with each other, 50 

Chapter III. — Constructions in Metal, 55 

Cage valve of a locomotive pump. Metallic pack- 
ing for stuffing-boxes, &c, .... 56 
Rolled-iron teams and columns, .... GO 

DIVISION III.— ELEMENTARY SHADOWS AND 
SHADING. 
Chapter I. — Shadows. 

§ I. — Facts, Principles, and Preliminary Problems, . 66 
§ II. — Practical Problems. (Twelve, with examples.) . 70 

Chapter II. — Shading, 77 

Hexagonal prism; cylinder; cone, sphere, and 
model. 

DIVISION IV.— ISOMETRIC AL AND OBLIQUE PROJEC- 
TIONS. 

Chapter I. — First Principles of Jsometrical Drawing, . . 87 

Chapter II. — Problems involving only Isometric Lines, . . 90 

Chapter III. — Problems involving Non-isometrical Lines, . . 95 
Chapter IV. — Problems involving the Constriction and Equal Division 

of Circles in Isometrical Drawing, ... 99 
Chapter V. — Oblique Projections, . . . . . .106 

DIVISION V.— ELEMENTS OF MACHINES. 

Chapter I. — Principles. Supporters and Crank Motions, . . 114 
Pillow-block; standard; bed and guide -rest; 
crank; ribbed eccentric; grooved eccentric. 

Chapter II. — Gearing, 126 

Spur-wheel ; bevel wheels ; screws and serpentines ; 
worm-wheel. 



CONTEXTS. V 

PAGE 

DIVISION VI.— SIMPLE STRUCTURES AND MACHINES. 

Chapter I. — Stone Structures, . . . . . . 140 

A brick segmental arch, 140 

A semi-cylindrical culvert, with vertical cylindri- 
cal wing- walls, 142 

Chapter II. — Wooden Structures, . . . .. . 146 

A king-post truss, 146 

A queen-post truss- bridge, . ■ t . . 147 
Chapter III. — Iron Constructions, . . . . . .151 

A railway track — frog, chair, &c, . . 151 

A hydraulic ram, 153 

Exercises. — A stop-valve; a Whipple truss-bridge; 
a vertical boiler; a Knowles steam-pump. 



NOTE TO THE FOEETH EDITION. 



This edition is improved chiefly by the extension of the chapter 
on oblique, or pictorial projections, also called mechanical per- 
spective, with the addition of a new plate. 

Numerous minor corrections, and new or improved paragraphs 
have been scattered through the work, as suggested by further ex- 
perience. Yet it is not expected that this edition, however im- 
proved, can supersede either careful and thinking study on the 
part of the willing student, or ample, repeated, and varied instruc- 
tion on the part of the willing teacher. Many minds require 
variously changed and amplified statements of the same thing 
before they are ready to exclaim heartily, "I see it now;" and the 
teacher must be ready to meet, with many forms of instruction, the 
conditions presented to him by different minds. 

A very moderate collection of such objects as are illustrated in 
Division II., and such as can be made by a carpenter or turner, or 
in machine, gas-fitting, or pattern shops, will very usefully supple- 
ment the text, and add to the pleasure and benefit derived by the 
student ; and will be better than an increased size of the book, 
which is meant to be rather suggestive than exhaustive, or to be 
too closely followed, in respect to the examples chosen for practice. 
Finally, the author's chief desire in relation to this, as well as the 
rest of his " elementary works," is, to see them so generally used in 
higher preparatory instruction as to give due place to higher 
studies in the same department in the strictly Technical Schools. 

R. P. L, Tboy, July, 1871. 



NOTE TO THE FIFTH EDITION. 



The call for a new edition of this manual has led the Author, 
after the lapse of ten years since the last revision, to make such 
further improvements in a new edition, as additional and varied 
experience and reflection have suggested. 

A few paragraphs m Division I. have been re-written. 

A few but valuable additions have been made to the text and 
plates of Div. II. 

Div. III. contains one more plate, taken from Div. VI., and 
examples of finished shading, not before provided. 

Div. IV. has beeu improved by the addition of a few figures 
and by partial re-writing. 

The most important change is the addition of Div. V., em- 
bracing the more important and universal elements of machines. 
The volume is thus made more complete, both in itself, and as an 
introduction to higher works both theoretical and practical.* 

Div. VI. (V. in previous editions) has been slightly enlarged 
by a few new and valuable practical examples. 

In general: while the subjects of all the Author's " Elementary 
Works " have been largely taught in the earlier classes of Poly- 
technic Schools, of whatever name, it is to be hoped that by the 
increasing development of scientific instruction, they will all 
ultimately be included in Preparatory Scientific Instruction, 
and in Special Normal Classes; and in behalf of the many pupils 
whose education ends in preparatory schools, but to whom an 
exact knowledge of elementary instrumental or mathematical 
drawing would be highly useful. 

The explanations of first principles have purposely been made 
very complete, in behalf of all classes of self-instructors, and 
because what is not thus printed must be said, and often repeated, 

* The beautiful plates XVII., XVIII., XIX., modified however to suit a 
full explanatory text, are from the Cours de dessin lineaire, par Delaistre, a 
work which every draftsman would do well to possess. 



Mil NOTE TO THE FIETH EDITION. 

to ensure that full understanding of the subject, the test of which 
is the ready performance of new examples. 

At this point the testimony of an evidently experienced and 
faithful English author and teacher may well be noted. Speak- 
ing of the copying system, he says, "If, however, at the end of 
one or two years' practice, the copyist [though able to make a 
highly finished copy] is asked to make a side and end elevation 
and longitudinal section of his lead-pencil, or a transverse section 
of his instrument-box, the chances are that he can do neither 
the one nor the other. Strange as it may appear, this is a state 
of things which I have had frequent opportunities for witness- 
ing. . . . The remedy has been to commence a course of 
study from the very beginning. ... He has made from the 
copy a highly finished drawing, with all the shadows admirably 
projected, being at the same time, however, perfectly ignorant 
of the rules for projecting such shadows. This is the true 
picture of a student who had a course of two years' study where 
mechanical drawing was taught " [by merely copying successively 
more and more elaborate drawings].* 

With these remarks the present edition, in its final form as 
now intended, is committed to the favor of Schools and Self- 
Instructors. 

Newton, Mass., October, 1880. 

* Preface to Binn's Orthographic Projections. London, 1867. 



FROM THE ORIGINAL PREFACE. 



Experience in teaching shows that correct conceptions of the 
forms of objects having three dimensions, are obtained with 
considerable difficulty by the beginner, from drawings having 
but two dimensions, especially when those drawings are neither 
"natural" — that is "pictorial" — nor shaded, so as to suggest 
their form ; but are artificial, or " conventional," and are merely 
" skeleton," or unshaded, line drawings. Hence moderate experi- 
ence suggests, and continued experience confirms, the propriety 
of interposing, between the easily understood drawings of pro- 
blems involving two dimensions, and the general course of pro- 
blems of three dimensions, a rudimentary course upon the methods 
of representing objects having three dimensions. 

Experience again proves, in respect to the drawing of any 
engineering structures that are worth drawing, that it is a great 
advantage to the draftsman to have — 1st, some knowledge of the 
thing to be drawn, aside from his knowledge of the methods of 
drawing it ; and 2d, practice in the leisurely study of the graphical 
construction of single members or elements of a piece of framing, 
or other structure. 

The truth of the second of the preceding remarks, is further 
apparent, from the fact that in entering at once upon the draw- 
ing of whole structures, three evils ensue, viz. — 1st, Confusion 
of ideas, arising from the mass of new objects (the many different 
parts of a structure) thrown upon the mind at once ; 2d, Loss 
of time, owing to repetition of the same detail many time* id 



X PREFACE. 

the same structure ; and 3d, Waste of drawings, as well as of 
time, through poor execution, which is due to insufficient pre- 
vious practice. Hence Divisions II. and V. contain a liberal 
collection of elements of structures and machines, each one of 
which affords a useful problem, while Division YI. includes 
examples of a few simple structures, to fulfil the threefold pur- 
pose of affording occasion for learning the names of parts of 
structures ; for practice in the combination of details into whole 
structures ; and for profitable review practice in execution. 

Classes will generally be found to take a lively interest in 
the subjects of this volume — because of their freshness to most 
learners, as new subjects of interesting study — because of the 
variety and brevity of the topics — and because of the compact- 
ness and beauty of the volume which is formed by binding to- 
gether all the plates of the course, when they are well executed. 
As to the use of this volume, it is intended that there should 
be formal interrogations upon the problems in the 1st, 3d, and 
4th divisions, with graphical constructions of a selection of the 
same or similar ones; and occasional interrogations mingled 
with the graphical constructions of the practical problems of 
the remaining divisions. Remembering that excellence in mere 
execution, though highly desirable and to be encouraged, is 
not, at this stage of the student's progress, the sole end to be 
attained, the student may, in place of a tedious course of 
finished drawings, be called on frequently to describe, by the 
aid of pencil or blackboard sketches, how he would construct 
drawings of certain objects — either those given in the several 
Divisions of this volume, or other similar ones proposed by his 
teacher. 



PRELIMINARY NOTES. 



As beginners not seldom find peciliar difficulties at the outset 
of the study of projections, the removal of which, however, 
makes subsequent progress easy, the following special explana- 
tions are here prefixed. 

I. Figs. 1, 2, 3, 5, and 15, of PI. I. are pictorial diagrams, 
used for illustration in place of actual models. Thus, in Fig. 3, for 
example, MHi represents a horizontal square cornered plane sur- 
face, as a floor. MGV represents a vertical square cornered 
plane surface, as a wall, which is therefore perpendicular to 
MIR P represents any point in the angular space included by 
these two planes. Pp represents a line from P, perpendicular to 
the plane MHi, and. meeting it at^>. T*p f represents a line from 
P, perpendicular to the plane MGY, and meeting it at_p'. Then 
Pp and Pp' are called the projecting lines of P. The point p is 
called the horizontal projection of P, and p' the vertical projection 
of P. The projecting lines of any point or of any body ) as in 
Fig. 1, are perpendicular to the planes, as MHi and MGrV, 
which are called planes of projection, 

II. In preparing a lesson from this work, the object of the 
student is, by no means, to commit to memory the figures, but 
to learn, from the first principles, and subsequent explanations, 
to see in these figures the realities in space which they represent; so 
as to be able, on hearing the enunciation of any of the problems, 
to solve it from a clear understanding of the subject, and not 
"by rote" from mere memory of the diagrams. The student 
will be greatly aided in so preparing his lessons, by working 
out the problems, in space, on actual planes at right angles to 
each other, as on the leaves of a folding slate, when one slate is 
placed horizontally and the other vertically. In the construction 
of his plates, he should also test his knowledge of the principles^ by 
varying the form of the examples, though without essentially 
changing their character. 



Xll 



PRELIMINARY NOTES. 



Drawing Instruments and Materials. 

This volume is intended to be the immediate successor of 
my " Drafting Instruments and Operations," which is there- 
fore supposed to have been read first, by students of this one. 

But as some self -instructors and other students may desire to 
acquire a knowledge of projections as quickly as possible, for 
practical use, and without regard, at first, to finished execution 
of their drawings, the following condensed information is here 
inserted for their convenience. 

To abridge the descriptions to the utmost, it may first be 
stated that dealers in Drawing Instruments and Materials are 
found in all large cities, who will send descriptive catalogues 
on application. Such are Frost & Adams, Boston ; "W. & 
L. E. Gurley, Troy, JS T . Y.; Keuffel & Esser, Xew York City; 
James "W. Queen, Philadelphia ; and others, doubtless, whose 
advertisements can be found in educational and popular me- 
chanical periodicals. The necessary articles are : — 

1. A good pair of compasses, with their furniture; that is, 
a pen, pencil, and needle point to replace the movable steel 
points, when drawing circles in pencil or ink. 

E= frr ^ . . mm 




2. A good drawing pen. 



3. A drawing board 20 x 30 inches 



A T square ; that is, a hard-wood ruler, having a stout 
wooden cross-piece about 2-J- x 9 inches, 
and half an inch thick, at one end. 
upon the flat side of which the blade 

is firmly fastened, truly at right angles. The blade may be 

about 30 inches loner. 



PRELIMINARY NOTES. 



Xlll 



5. A pair of hard-wood right-angled triangles, the longest 
side about 10 inches long ; one with the two acute angles of 
45° each, the other with acute angles of 30° and 60°. 




6. A triangular scale, graduated into tenths or twelfths of 
the unit as may be preferred ; or, a flat ivory " protractor 
scale." 






7. Buff manilla ofhce, or "detail" paper, or, if preferred and 
it can be afforded, Whatman's rough drawing paper, of con- 
venient size, from "medium," 17" x 22", to "imperial," 
21" x 30". 

8. Hard lead-pencils. 

9. A cake of Indian ink — Chinese the best for shading, 
the Japanese for lines. 

Where the utmost economy is sought, a very cheap, fair 
quality of brass instruments can be had in boxes, or the draw- 
ings can even be made with pencil only ; any neat worker in 
hard wood can make the drawing board, T square, and triangles, 
and a foot-rule may be made to serve as a scale. 

When drawings are not to be colored, the paper can be lightly 
gummed or tacked to the drawing board at the corners. Other- 
wise, the sheet should be well wet by sponging with clean 
water and, while wet, fastened to the board by means of thick 
mucilage applied along the edge of the paper. 

Indian ink is prepared for use by rubbing it on a bit of 
china, with a few drops of water. It is then applied between 
the blades of the drawing pen by a small feather or slip of 
paper. Pen and ink should be wiped dry when done with. 



ELEMENTARY PROJECTION DRAWING. 



DIVISION FIRST. 

PROJECTIONS. 



CHAPTER I. 

FIRST PRINCIPLES. 

§ 1 . TJie purely Geometrical or Rational Theory of Projections. 

1. Elementary Projection Drawing is an introduction tc 
Descriptive Geometry, and shows how to represent simple solids. 
singly and in combination, upon plane surfaces, yet so as to show 
their real dimensions. 

2. If ten feet of 5-inch stove-pipe were wanted, a circle five 
inches in diameter, drawn on paper, would be all the pattern the 
workman would need. But if the desired length were forgotten, 
or if the pipe were to be conical, the circular drawing would no 
longer be sufficient. That is, as a plane surface has but two di- 
mensions, no more than two dimensions of any object can be ex- 
actly shown in one figure on that plane. 

But practical ivork, and geometrical problems for study, are 
both continually arising, which require, for convenient execution 
in one case, ®oA proper solution in the other, that we should be 
able, in some way, to truly show all the dimensions of solid 
bodies upon plane surfaces. 

3. What, then, is the number and the relative position of the 
planes which will enable us to represent all the dimensions of any 
geometrical solid, in their real size, on those planes? To assist in an- 
swering this question, reference maybe made to PL I., Fig. 1. Let 
ABCFED be a regular square-cornered block, whose length is AB; 
breadth, AD ; and thickness, AC ; and let MN be any horizontal 
plane below it and parallel to its top surface ABED. If now from 
the four points A, B, D and E, perpendiculars be let fall upon the 
horizontal plane MN, they will meet it m the points a, b, d and e. 



2 FIRST PRINCIPLE.-. 

By joining these points, it is evident that a figure — abed — will be 
formed, which will be equal to the top surface of the block, and 
will be a correct representation of the length and breadth of that 
top surface — i. e. of the length and breadth of the block. Simi- 
larly, if MP be a vertical plane, parallel to the front, ABCF, of 
the block; and if perpendiculars, Aa' , etc., be let fall from 
A, B, C, F, upon MP; the figure, a'b'c'f, will be equal to ABCF, 
and hence will show the length and thickness of the block. 

4. From the last article the following definitions arise. The 
point a, PI. I., Fig. 1, is where A would arrive if thrown, that is, 
projected, vertically downwards along the line Aa. Likewise, a' 
is where A would be, if thrown or projected along Aa', from A 
to a', perpendicularly to the plane MP. Hence a is called a 
horizontal projection of A ; and a' is called its vertical projection. 

Also, conversely, A is said to be horizontally projected at a, 
and vertically projected at a'. 

The plane MX is thence called the horizontal plane of projec- 
tion ; and the plane MP, the vertical plane of projection. 

Aa and Aa' are called the two projecting lines of the point A 
relative to the planes of projection, MX and MP. 

Hence we have this definition. If from any given point a per- 
pendicular be let fall upon a plane, the point where that perpen- 
dicular meets the plane will be the projection of the point upon the 
plane, and the perpendicular will be the projecting line of the point. 

5. Similar remarks apply to any number of points or to objects 
limited by such points; and to their projections uj:>on anv other 
planes of projection. Thus (Coed is the horizontal projection of 
the block ABCD, and a'b'c'f is its vertical projection, and, 
generally, the projecting lines of objects are perpendicular to the 
planes of projection employed. Finally, the intersection, as MR, 
of a horizontal, and a vertical plane of projection, is the ground 
line for that vertical plane. 

6. From the foregoing articles the following principles arise. 
First: Two planes, at right angles to each other, are necessary 
to enable us to represent, fully, the three dimensions of a solid. 
Second: In order that those dimensions shall be seen in their true 
size and relative position, they must be parallel to that plane on 
which they are shown. Third: Each plane shows two of the di- 



FIRST PRINCIPLES. 3 

mensions of the solid, viz., the two which are parallel to it; and 
that dimension which is thus shown twice, is the one which is 
parallel to both of the planes. Thus AB, the length, and AD, 
the breadth, are shown on the plane MN ; and AB, the length 
again, and AC, the thickness, are shown on the plane MP, 
Fourth : The height of the vertical projection of a point above 
the ground line, is equal to the height of the point itself, in 
space, above the horizontal plane; and the perpendicular distance 
of the horizontal projection of a point from the ground line, is 
equal to the perpeyidicular distance of the point itself, in front 
of the vertical plane. Thus : PL L, Fig. 1, aa" = Ka' and 
a'a" = Aa, 

7. The preceding principles and definitions are the foundation ol 
the subject of projections, but, by attending carefully to PI. I., Fig. 
2, some useful elementary applications of them may be discovered, 
which are frequently applied in practice. PI. I., Fig. 2, is a pic- 
torial model of a pyramid, Ycdeg, and of its two projections. The 
face, Vcd, of the pyramid, is parallel to the vertical plane, and the 
triangle, Xa#, is equal and parallel to Vcd, and a little in front and 
at one side of it. By first conceiving, now, of the actual models, 
which are, perhaps, represented as clearly as they can be by mere 
diagrams, in PL I., Figs. 1 and 2 ; and then by attentive study of 
those figures, the next two articles may be easily understood. 

§ II. — Of the Relations of Lines to their Projections. 

8. Relations of single lines to their projections. 

a. A vertical line, as AC, PI. I., Fig. 1, has, for its horizontal 
projection, a point, a, and for its vertical projection, a line a'c\ 
perpendicular to the ground line, and equal and parallel to the line 
AC, in space. 

b. A horizontal line, as AD, which is perpendicular to the verti- 
cal plane, has, for its horizontal projection, a line, ad, perpen- 
dicular to the ground line, and equal and parallel to the line, AD, 
in space ; and for its vertical projection a point, a'. 

c. A horizontal line, as AB, which is parallel to both planes of 
projection, has, for both of its projections, lines ab and a'b', which 
are parallel to the ground line, and equal and parallel to the line, 
AB, in space. 

d. A horizontal line, as BD, which makes an acute angle with 
the vertical plane, has, for its horizontal projection, a line, bd, 
which makes the same angle with the ground line that the line, BD. 



FIRST PRINCIPLES, 



makes with the vertical plane, and is equal and parallel to the line 
itself (BD) ; and has for its vertical projection a line b'd, which is 
parallel to the ground line, but shorter than BD, the line in space. 

e. An oblique line, as BC, PI. I., Fig. 1, or Yd, PI. I., Fig. 2, 
which is parallel to the vertical plane, has, for its vertical projection, 
a line b'c', or v'd', which is equal and parallel to itself, and for its 
horizontal projection, a line ba or vd, parallel to the ground line, 
but shorter than the line in space. 

/. An oblique line, as Yg, PI. I., Fig. 2, which is oblique to both 
planes of projection, has both of its projections, v'd' and vg, oblique 
to the ground line, and shorter than the line itself. 

g. An oblique line, as AH, PI. I., Fig. 1, which is oblique to both 
planes of projection, but is in a plane ACDH, perpendicular to 
both of those planes, has both of its projections, a'c' and ad, perpen- 
dicular to the ground line, and shorter than the line itself. 

h. A line, lying in either plane of projection, coincides with its 
projection on that plane, and has its other projection in the ground 
line. See cd — c'd', the projections of cd, PI. I., Fig. 2. 

9. Remark. A general principle, which it is important to be 
perfectly familiar with, is embodied in several of the preceding 
examples; viz. When any line is parallel to either plane of projec- 
tion, its projection on that plane is equal and parallel to itself, and 
its projection on the other plane is parallel to the ground line. 

10. The preceding remark serves to show how to find the true 
length of a line, when its projections are given. When the line, as 
Yg, PI. I., Fig. 2, is oblique to both planes of projection, its length, 
Yg, is evidently equal to the hypothenuse of a right-angled triangle, 
of which the base is vg, the horizontal projection of the line, and the 
altitude is Yv, the height of the upper extremity, Y, above the 
horizontal plane. When the line, as AH, PI. I., Fig. 1, does not 
touch either plane of projection, it is evidently equal to the hypo- 
thenuse of a right-angled triangle, of which the base, CH, equals 
the horizontal projection, ad, and the altitude, AC, equals the 
difference of the perpendiculars, Aa and Hd, to the horizontal plane. 

In the same way, it is also true that the line, as Yg, PI. I., Fig. 2, 
is the hypothenuse of another right-angled triangle, whose base 
equals the vertical projection, v'd', and whose altitude equals the 
difference of the perpendiculars, Yv' and d'g, from the extremities, 
of the line to the vertical plane of projection. 

11. Relations of pairs of lines to their projections. These rela* 
tions, after the full notice now given of the various positions of 
single lines, may be briefly expressed as follows. 



FIRST PRINCIPLES. I 

a. A pair of lines which are equal and parallel in space, a?idalsa 
parallel to a plane of projection, as AB and CF, PL I., Fig. 1, orVc 
and Xa, PI. I., Fig. 2, have their projections on that plane — a'b' 
and c'f, PI. I., Fig. 1, or v'c' and x'a\ PI. I., Fig. 2 — equal and 
parallel — to each other, and to the lines in space. 

b. A pair of lines which are equal and parallel in space, but not 
parallel to a plane of projection, will have their projections on that 
plane equal and parallel to each other, but not to the lines in 
space. 

c. Parallel lines make equal angles with either plane of projec- 
tion ; hence it is easy to see that lines not parallel to each other — 
as Yd and Vc, or Yg and Ye, PI. I., Fig. 2 — but which make 
equal angles with the planes of projection, will have equal projec- 
tions on both planes — i.e. v'd'—v'c' and vg~ve, also vd=vc. 

§ III. — Physical Theory of Projections. 

12. The preceding articles comprise the substance of the purely 
geometrical or rational theory of projections, which, strictly, is 
sufficient ; but it is natural to take account of the physical fact that 
the magnitudes in space and their representations, both address 
themselves to the eye, and to inquire from lohat distance and in 
what direction the magnitudes in PI. I., Figs. 1 and 2, must be 
viewed, in order that they shall appear just as their projections 
represent them. Since the projecting lines, Q, regarded as rays, 
reflected from the block, Fig. 1, to the eye, are parallel, they could 
only meet the eye at an infinite distance in front of the vertical 
plane. Hence the vertical projection of an object represents it as it 
would appear to the eye, situated at an infinite distance from it, and 
looking in a direction perpendicular to the vertical plane of pro- 
jection. Likewise, the projecting lines, S, show that the horizontal 
projection of an object represents it as seen from an infinite distance 
above it, and looking perpendicularly down upon the horizontal 
plane. Thus, the projecting lines represent the direction of vision^ 
which is perpendicular to the plane of projection considered. 

§ IV. — Conventional Mode of representing the two Planes of Pro- 
jection, and the two Projections of any Object upon one plant 
— viz. the Plane of the Paper. • 

13. In practice, a single flat sheet of paper represents the two 
planes of projection, and in the following manner. The vertical 
plane, MV, PL I., Fig. 3, is supposed to revolve backwards, as 



b PIRST PRINCIPLES. 

shown by the arcs ru and Vt, till it coincides with the horizontal 
plane produced atM.utGc. Hence, drawing a line from right to 
left across the paper, to represent the ground line, MG, all that 
part of the paper above or beyond such a line will represent the 
vertical plane of projection, and the part below it the horizontal 
plane of projection. 

14. Elementary geometry shows that the plane, as PP' p"p, 
PI. I., Fig. 3, of the projecting lines, Vp and PP', (3, 4) is per- 
pendicular to both of the planes of projection, and to the ground 
line MG. Hence it intersects these planes in lines, as pp" and 
P' p" , both of which are perpendicular to the ground line at the 
same pointy". 

15. If, now, as explained in (13) the vertical plane MV, PL I., 
Pig. 3, be revolved about MG, to coincide with the horizontal 
plane, the point p" will remain in the axis MG, and the lines p'p" 
and Py will unite to form one line pp f , perpendicular to MG. 

That is: Whenever tiuo points are the projections of one point 
in space, the line joining them will be perpendicular to the ground 
line. 

§ V. — Of the Conventional Direction of the Light ; and of the 
Position and Use of Heavy Lines. 

10. Without going into this subject fully, as in Div. III., it is 
sufficient to say here that, as one faces the vertical plane of pro- 
jection, the light is assumed to come from behind, and over the 
left shoulder, in such a direction that each projection of a ray 
(but not the ray itself) makes an angle of '45° with the ground line, 
as shown in PL I., Fig. 6. And note that the light is supposed 
to turn with the observer, as he turns to face any other vertical 
]}lane. 

17. The practical effect of the preceding assumption in refer- 
ence to the light, is, that upon a body of the form and position 
shown in PL I., Fig. 5, for example, the top, front, and left 
hand surfaces — i. e. the three seen in the Fig. — are illuminated, 
while the other three faces of the bodv are in the shade. 



FIRST PRINCIPLES. 7 

18. The practical rule by which the direction of the light, and 
its effect, are indicated in the projections, is, that all those visible 
edges of the body in space, which divide the light from the dark 
surfaces, are made heavy in projection. 

19. To illustrate : The edges BC and CD of the body in space, 
PI. I., Fig. 5, divide light from dark surfaces, and are seen in 
looking towards the vertical, plane, and hence ar6 made heavy in 
vertical projection, as seen at b'c' and c'd'. BK and KF divide 
illuminated from dark surfaces, and are seen in looking towards 
the horizontal plane, and are therefore made heavy in the hori- 
zontal projection, as shown at bh and kf. 

20. By inspection, it will be seen that the following simple rule 
in reference to the position of the heavy lines on the drawings, may 
be deduced, as an aid to the memory. In all ordinary four-sided 
prismatic bodies, placed with their edges respectively parallel and 
perpendicular to the planes of projection, or nearly so, the right 
hand lines, and those nearest the ground line, of both projections, 
are made heavy. 

21. Heavy lines are of considerable use, in the case of line draw- 
ings particularly, in indicating the forms of bodies, as will be seen 
in future examples. In shaded drawings, the student must be 
careful to omit the heavy, or " shade lines," which habit, in mak- 
ing many line drawings, might lead him to add. On flat colored 
surfaces they should be added last, to avoid washing them, when 
■coloring. 

§ VL— Notation. 

22. Under the head of Notation, two points are to be consi- 
dered, the manner of indicating the various lines of the diagram, 
and the lettering. As will be seen by examining PI. I., Figs. 1, 2 
— see Ye, eg, &c. — and 5, the visible lines of the object represented 
are indicated by full lines ; lines of construction and invisible lines 
of the object, so far as they are shown, are made in dotted lines. 
The intersections of auxiliary planes with the planes of projection, 
called traces, are represented by broken and dotted lines, as at 
PQP', PI. I., Fig. 16. 

23. Unaccented letters indicate the horizontal projections of 
points. The same, with one or more accents, denote their vertical 
projections. The simple rule of thus always lettering the same 
point with the same letter, wherever it is shown, affords a key to 
every diagram, as will be shown as the course proceeds. 

The projections of a body are geometrically equivalent to the 



O FIRST PRINCIPLES. 

body itself, since they show its form, position and dimensions. 
Hence objects are considered as named by naming their projections. 
Thus, the point pp' means the point whose projections are p &ndp'; 
the line ab — a'V means the one whose projections oreab and a'V '. 
For brevity, the horizontal and vertical planes of projection are 
designated, respectively, as H and V. 

24. In the practical applications of projections, "horizontal pio» 
jections" are usually called "plans" and "vertical projections," 
" elevations?"* 

Before entering upon the study of the subsequent constructions, 
the terms "perpendicular''' and " vertical" should be clearly dis- 
tinguished. " Perpendicular" is a relative term, showing that any 
line or surface, to which it is applied, is at right angles to some 
other line or surface. " Vertical" is an absolute term, at any one 
place, and applies to any line or surface at right angles to a level, 
as a water surface. A vertical line, L, is perpendicular to all hori- 
zontal lines which intersect it, but if the entire system of lines thus 
related were inclined, so that all should be oblique, L would still 
be perpendicular to all the rest, though no longer vertical. 

§ VII. — Of the Use of the Method of Projections. 

25. Under this head it is to be noticed, that all drawings are 
made to serve one or the other of two purposes, i.e. they are made 
for use in aiding workmen in the construction of works ; or in 
rendering intelligible, by means of drawings, the real form and size 
of some existing structure ; or else, they are made for ornament, 
or to embellish our houses and gratify our tastes, and to show the 
apparent forms and relative sizes of objects. 

26. Drawings of the former kind are often called, on account of 
the uses to which they are applied, "mechanical" or "working" 
drawings. Those of the latter kind are commonly called pictures ; 
and here it is to be noticed that if " working " drawings are to 
show the true, and not the apparent, proportions of all parts of an 
object, they must, all and always, conform to this one rule, viz. 
All those lines which are equal and similarly situated on the object, 
must be equal and similarly situated on the drawing. 

But, as is now abundantly evident, drawings made according to 
the method of projections, do conform to this rule; hence their use r 
as above described. 



CHAPTER IL 

PROJECTIONS OF LINES: PROBLEMS IN RIGHT PROJECTION; AND 
INCLUDING PROJECTIONS SHOWING TWO SIDES OF A SOLID 
RIGHT ANGLE. 

27. The style of execution of the following problems is so simple, 
and so nearly alike for all of them, that it need not be described 
for each problem separately, but will be noticed from time to time. 
In the solution of problems, lines are considered as unlimited, and 
may be produced indefinitely in either direction, 

§ I. — Projections of Straight Lines. 

28. Prob. 1. To construct the projections of a vertical straight 
line, 1^- inches long, whose lowest point is % an inch from the 
horizontal plane, and all of whose points are % of an inch from 
the vertical plane. 

Remarks, a. The remaining figures of PI. I. are drawn just 
half the size indicated by the dimensions given in the text. It 
may be well for the student to make them of full size. 

b. Let MG be understood to be the ground line for all of the 
above problems, without further mention of it. 

1st. Draw, very lightly, an indefinite line perpendicular to the 
ground line, PI. I., Fig. 7. 

2d. Upon it mark a point, a', two inches above the ground line, 
and another point, b', half an inch above the ground line. 

3d. Upon the same line, mark the point a,b, three-fourths of an 
inch below the ground line. Then a' b' will be the vertical, and 
ab the horizontal projection of the required line. (8 a) 

29. Prob. 2. To construct the projections of a horizontal line, 
1-^ inches long, \\ inches above the horizontal plane, perpendicular 
Co the vertical plane, and with its furthermost point— from the eye 
— \ of an inch from that plane. PI. I., Fig. 8, in connection with 
the full description of the preceding problem, will afford a sufficient 
explanation of this one. 

JRemarJc. It often happens that a diagram is made more inte) 



19 PROJECTIONS OF LJXEb 

ligible by lettering it as at ab, PL I., Fig. % and at c'd\ PI. I. Fig 
8, for thus the notation shows unmistakably, that ab or c'd' are not 
the projections of points but of lines. 

30. Problems 3 to 8, inclusive, need now only to be enunciated, 
with references to their constructions, in PI. I. 

Fig. 9 shows the projections of a line, 2 J inches long, parallel to 
the ground line, 1|- inches from the horizontal plane, and 1 inch 
from the vertical plane. 

Fig. 10 is the representation of a line, 2 inches long ; parallel to 
the horizontal plane, and 1 inch above it ; and making an angle of 
30° with the vertical plane. 

Fig. 11 represents a line, 2^ inches long, parallel to the vertical 
plane, and lj inches from it, and making an angle of 60° with the 
horizontal plane. 

Fig. 12 gives the projections of a line, 1^- inches long, lying in 
the horizontal plane, parallel to the ground line, and l£ inches 
from it. The projection a'~b r shows the line to be in H (6, 4th). 

Fig. 13 shows the projections of a line, lj inches long, lying in 
the vertical plane, parallel to the ground line, and 1 inch from it. 

Fig. 14 indicates a line, 2\ inches long, lying in the vertical 
plane, and making an angle of 60° with the horizontal plane. 

31. Projections of the revolution of a point about an axi 
When a point revolves about an axis, it describes a circle, or arc, 
whose plane is perpendicular to the axis. Thus a point, revolving 
about an axis which is perpendicular to the vertical plane, describes 
an arc, parallel to that plane. The vertical projection of such an 
arc is an equal arc. Its horizontal projection (6) is a straight line 
parallel to the ground line. 

Thus, PI. I., Fig. 15a, Ga represents a perpendicular to the ver 
tical plane, Y. The point, A, by revolving a certain distance about 
this axis, describes the arc AB; whose vertical projection is the 
equal arc, a'b' '; and whose horizontal projection is ab, a straight 
line parallel to the ground line. 

Likewise, briefly, in Figs. 15b and 15c, XY is a vertical axis. 
The point A, revolving about it, describes a horizontal arc, AB; 
whose horizontal projection, db, is an equal arc ; and whose verti- 
cal projection, a'b\ is a straight line parallel to the ground line. 

32. Prob. 9. To construct the projections of a line which is in a 
plane perpendicular to both planes of projection, the line being 
oblique to both planes of projection. Plate I., Fig. 15, represents a 



PROJECTION OF LINES. 1] 

model of this problem. AB represents the line in 8pace ; ab its 
horizontal projection; a'b' its projection on the vertical plane MP'; 
and A'B' its projection on an auxiliary vertical plane PQP'; which 
is parallel to AB, and perpendicular to the ground line. Hence 
A'B'=AB. 

Now in making these three planes of projection coincide with 
the paper, taken as the horizontal plane of projection, the plane 
PQP' is revolved about P'Q as an axis, till it coincides with the 
primitive vertical plane, MP', produced, as at P'QV", and then the 
united vertical planes, MP'V", are revolved backward about MH' 
as an axis into the horizontal plane. In the first revolution, A' 
describes, according to the last article, the horizontal arc, A'a", 
(31) about m as a centre, and whose projections are a""a'", having 
its centre at Q, and ma!'. Also B' describes the horizontal arc, 
B'b", about n as a centre, and whose projections are b""b'" , whose 
centre is Q, and rib". Thus we see that two or more different ver- 
tical projections, as a' and a", of the same point, are in the same 
parallel, a'a", to the ground line; that is, they are at the same 
height above that line. Hence a" is at the intersection of a'a", 
parallel to the ground line, MH 7 , with a'" a", perpendicular to MH'. 

33. a. Notice further that a""b"" is the horizontal projection of 
A'B', and that it coincides with the projection of ab upon PQP' 
Likewise, that mn is the vertical projection of A'B', and that it co 
incides with the projection of a'b' upon PQP'. 

b. Note that Bb', for example (6), is equal to bt, and that bt=b"n 
the distance of the auxiliary vertical projection, b", of B, from the 
trace, or axis, P'Q, of the auxiliary plane. 

c. Note that a"b" shows the true length and direction of AB ; 
that is, the angles made by a"b" with H'Q and P'Q, respectively, 
are equal to those made by AB with the planes of projection. 

34. To construct PI. I., Fig. 15, in projection. See PI. I, Fig. 
16, where, to make the comparison easier, like points have the 
same letters as in Fig. 15. Supposing the length and direction of 
the line given, we begin with a"b", which suppose to be 2" long, 
and to make an angle of 60° with the horizontal plane. Suppose 
the line in space to be 1-J- inches to the left of the auxiliary vertical 
plane P'QP then a' b', its vertical projection, will be perpendicular 
to the ground line, between the parallels a" a' and b"b' (32), and 1£ 
inches from P'Q. The horizontal projection, ab, will be in a'b' 
produced ; b"n — b'"b"" are the two projections of the arc in which 
the point b" revolves back to its position, n — b"", in the plane 
P'QP, and b""b — rib' is the line in which nb"" is projected back 



12 PROJECTION OF LINES. 

to its primitive position b'b. Therefore, b is at the intersection 
oib""b with a'b' produced, a is similarly found, giving ab as the 
horizontal projection of the given line. 

Art. 32 shows sufficiently how to find the length a"b f, \iab — a'b* 
were given. 
-Example. Construct the figure when a',b is the highest point. 

35. Execution. The foregoing problems are to be inked with 
very black ink; the projections of given lines, and the ground line, 
in heavy full lines ; and the lines of construction mfine dotted lines 
as shown in the figures. Lettering is not necessary, except for 
purposes of reference, as in a text book, though it affords occasion 
for practice in making small letters. 

On the other hand, lettering, if poorly executed, disfigures a 
diagram so much that it should be made only after some previous) 
practice, and then carefully; making the letters small, fine, and 
regular. 

§ II. — Bight Projections of Solids. 

Remark. The term " right projection " becomes significant only 
when it refers to bodies which are, to a considerable extent, 
bounded by straight lines at right angles to each other. Such 
bodies are said to be drawn in right projection when their most 
important lines, and fices, are parallel or perpendicular to one or 
the other of the planes of projection. 

36. Pkob. 10. — To construct the p>rojections of a vertical right 
prism, having a square base ; standing upon the horizontal plane, 
and with one of its faces parallel to the vertical plane. PI. II., 
Fig. 17. 

Let the prism be 1 inch square, \\ inches high, and \ of an inch 
from the vertical plane. 

\st. The square ABEF, \ of an inch from the ground line, is the 
plan of the prism, and strictly represents its upper base. 

2(7. A'B'C'D', \\ inches high, is the elevation of the prism, and 
strictly represents its front face. 

In this, and in all similar problems, it is useful to distinguish the 
positions of the points, lines, and faces, in words ; as upper and 
lower ; front and back ; right and left ; just as is done in speaking 
of the bodies which (23) the projections represent. Thus, 

1st. AA' is the front, upper, left hand corner of the prism. 

2c?. EF— A'B' is the back top edge ; BF— D' is the lower right 
hand edge/ each corn^ of the plan is the horizontal projection of 
a, vertical edge ; etc. 



PROJECTIONS OP SOLIDS. 13 

3d. AE — A'C is the left hand face; etc. 

37. Prob. 11. — To construct the plan and two elevations of a 
vrism having the proportions of a bride, and placed with its length 
parallel to the ground line. Plate II., Fig. 18. 

1st. abed is the plan, f of an inch broad, twice that distance in 
length, and § of an inch from the ground line, showing that the prism 
in space is at the same distance from the vertical plane of projection. 

2nd. a'b'e'f is the elevation, f of an inch thick, and as long 
as the plan ; and J of an inch above the ground line, showing that 
the prism in space is at this height above the horizontal plane. 

Zrd. If a plane, P'QP, be placed perpendicular to both of the 
principal planes of projection, and touching the right hand end 
of the prism, it is evident that the projection of the prism upon 
such a plane w r ill be a rectangle, equal, in length, to the width, bd, 
of the plan, and, in height, to the height, b'f of the side elevation. 
This new projection will also, evidently, be at a distance from the 
primitive vertical plane, i.e. from P'Q, equal to dQ, and at a dis- 
tance from the horizontal plane equal to Qf. When, therefore, 
the auxiliary plane, P'QP, is revolved about P'Q into the primitive 
vertical plane of projection, the new projection will appear at 
a"e"c"g". 

4th. dc'" is the horizontal, and b'c" the vertical projection of 
the arc in which the point db' revolves into the primitive vertical 
plane. ba"\ b'a", are the two projections of the horizontal arc m 
which the corner bb' of the prism revolves. 

Example. — Let the auxiliary plane PQP' be revolved about PQ 
into the horizontal plane. a"c" will then appear to the right of 
PQ and at a distance from it equal to Q6'. 

38. Prob. 12. — To construct the two projections of a cylinder 
which stands upon the horizontal plane. PI. II., Fig. 19. 

The circle AaB5 is evidently the plan of such a cylinder, and 
the rectangle A'B'C'D'its elevation. Observe, here, that while the 
elevation, alone, is the same as that of a prism of the same height, 
Fig. 17, the plan shows the body represented, to be a cylinder. 

Any point as a in the plan, is the horizontal projection of a ver 
deal line lying on the convex surface, and called an element. A— 
A'C, and B — B'D', which limit the visible part of the convex sur- 
face, are called the extreme elements. 

39. As regards execution, the right hand line B'D' of a cylinder 



14 PROJECTIONS OF SOLIDS. 

or cone may be made less heavy than the line B'D', Fig. 17 ; and 
in the plan, the semicircle, aBb, convex towards the ground line, 
and limited by a diameter ab, which makes an angle of 45° with 
the ground line, is made heavy, but gradually tapered, into a fine 
line in the vicinity of the points a and b. 

40. Prob. 13. — To construct the projections of a cylinoZer whose 
axis is placed parallel to the ground line. PI. II., Fig. 20. 

Let the cylinder be \\ inches long, £ of an inch in diameter, its 
axis £ of an inch from the horizontal plane, and \ an inch from the 
vertical plaDe. The principal projections will, of course, be two 
equal rectangles, gehf and a'b'c'd', since all the diameters of the 
cylinder are equal. The centre lines, g'h f and ab, are made at the 
same distances from the ground line, that the axis of the cylinder 
is from the planes of projection (6). 

The end elevation, knowing its radius, which is equal to half of 
the diameter ye, or a'c f , of the cylinder, may be made by revolving 
the projection of its centre a,g f , only, upon PQP', around P'Q as 
an axis. 

41. Standing with a horizontal cylinder before one, with its axis 
lying from right to left, and parallel to the ground line, one of its 
elements is its highest one, that is the highest above the ground; or 
the horizontal plane ; another is the lowest ; another, the foremost, 
that is the one nearest to one, and another the hindmost, or the one 
furthest from one. Transfering the same terms to the projections 
of the same elements, by (23) we have ab — a' b' — b" [the three pro- 
jections of] the highest element; ab — c'd' — d", the lowest element; 
gh — g'h' — A", the foremost element ; and ef — g'h'—f", the hind- 
most element. 

In inking, the end elevation, b"f"d", is made heavy at nf"d"p, 
and tapered into a fine line in the vicinity of n and^y because, by 
(16) when the observer turns to face the plane PQP', looking at it 
hi the direction hg (12), the light turns with him. 

42. We have now three ways of distinguishing the projections 
of a horizontal cylinder from those of a square prism of equal 
dimensions. First, by medium instead of fully heavy lines od 
ef and </$. Second, by the lettering of the principal elements, av 
just explained. Third, and most clearly, by the circular end ele 
vation. 

§ III — Projections showing two sides of a Solid Right Angle. 
43 A solid right angle is an angle such as that at any corner of 



PROJECTIONS OP SOLIDS. 15 

a cube, or a square prism, and is therefore bounded by three plane 
right angles. When two faces of such a body are seen at once 
they will be seen obliquely, and neither will appear in its true size. 
Hence only one of the projections of the object will show two of 
its dimensions in their real size. Hence, we must always make, 
first, that projection, whichever it be, which shows two dimensions 
in their real size. 

44. Peob. 14. — To construct the plan and two elevations of a 
vertical prism, with a square base; resting on the horizontal plane, 
and having its vertical faces inclined to the vertical plane of pro- 
jection. PI. H., Figs. 21 — 22. 

1st. ABCG is the plan, which must be made first (43) and with 
its sides placed at any convenient angle with the ground line. 

2d. A'B'D'E' is the vertical projection of that vertical face 
whose horizontal projection is AB. 

3d. B'C E'F' is the vertical projection of that face whose hori- 
zontal projection is BC. This completes the vertical projection 
of the visible parts of the prism, when we look at the prism in the 
direction of the lines CF', &c. 

4th. Let gb be the horizontal trace of an auxiliary vertical plane 
of projection, which is perpendicular to both of the principal planes 
of projection. In looking perpendicularly towards this plane, i.e. 
in the directions Gg, &c, AG and AB are evidently the horizontal 
projections of those vertical faces that would then be visible; and 
the projecting lines, Gg, Aa, and Bb determine the widths ga and 
ab of those faces as seen in the new elevation. Now the auxiliary 
plane gb is not necessarily revolved about its vertical trace (not 
shown), but may just as well be taken up and transferred to any 
position where it will coincide with the primitive vertical plane ; 
only its ground line gb must be made to coincide with the principal 
ground line, as at H'E". Hence, making H'D" and D"E" respec- 
tively equal to ga and ab, and by drawing H"G", &c., the new 
elevation will be completed. 

45. The two elevations — PI. n., Figs. 21, 22 — appear exactly 
alike, but the faces seen in Fig. 22 are not the same as the equa. 
ones of Fig. 21. 

The different projections of the same face may be distinguished 
by marks. Thus the surfaces marked fl: are the two elevations of 
the same face of the prism ; the one marked <£ is visible only on the 
first elevation, and the one marked x is visible only on the second 
elevation — Fig. 22. 



16 PROJECTIONS OF SOLIDS. 

46. PI. II., Fig. 23, represents a small quadrangular prism in two 
elevations, the axis being horizontal in space, so that the left hand 
elevation shows the base of the prism. In the practical applica- 
tions of this construction, the centre, s, of the square projection is 
generally on a given line, not parallel to the sides of the square. 
Hence this construction affords occasion for an application of the 
problem : To draw a square of given size, with its centre on a given 
line, and its sides not parallel to that line. The following solution 
should be carefully remembered, it being of frequent application. 
Through the given centre, s, draw a line, L, in any direction, and 
another, L', also through s, at right angles to L. On each of these 
lines, lay off each way from s, half the length of a side of the 
square. Through the points thus formed, draw lines parallel to 
the lines L and 1/ and they will form the required square whose 
centre is s. 

47. Peob. 15. — lb construct the plan and several elevations of a 
vertical hexagonal prism, which rests upon the horizontal plane oj 
projection. PI. II., Figs. 24, 25, 26. 

The distinction between bodies as seen perpendicularly, or ob- 
liquely, becomes obscure as we pass from the consideration of 
bodies whose surfaces are at right angles to each other. Figs. 24 
and 25 show a hexagonal prism as much in right projection as such 
a body can be thus shown, but, as in both cases a majority of its 
surfaces are, considered separately, seen obliquely, its construction 
is given here. 

In Fig. 24 the hexagonal prism is, as shown by the plan, placed 
so that two of its vertical faces are parallel to the vertical plane 
of projection. Observe that where the hexagon is thus placed, 
three of its faces wiD be visible, one of them in its real size, viz., 
BC, B'C'F'G', and that the extreme width, E'H', of the eleva- 
tion, equals the diameter, AD, of the circumscribing circle of the 
plan. This is therefore the widest possible elevation of this prism. 
Notice, also, that as BC equals half of AD, while AB and CD are 
equal, and equally inclined to the vertical plane, the elevations, A'F' 
and G'D', of these latter faces, will be equal, and each half as wide 
as the middle face. This fact enables us to construct the elevation 
of a hexagonal prism situated as here described, without construct- 
ing the plan, provided w r e know the width and height of one face 
of the prism. This last construction should be remembered, it 
being of frequent and convenient application in the drawing of 
•outs, bolt-heads, &g., in machine drawing. 



PBOJECT70XS OF SOLIDS. 17 

48. PI. It., Fig. 25 shows the elevation of the same- prism on a 
plane which originally was placed at ib, and perpendicular to the 
horizontal plane ; whence it appears, that if a certain elevation of a 
hexagonal prism shows three of its faces, and one of them in its 
full size, another elevation, at right angles to this one, will show 
but two faces, neither of them in its full size ; the extreme width. 
I"B", of the second elevation being equal to the diameter of the 
inscribed circle of the plan. This is therefore the narrowest pos- 
sible elevation of this prism. 

49. PL II., Fig. 26 shows the elevation of the same prism as it 
appears when projected upon a vertical plane standing on jb", and 
then transferred to the principal vertical plane, at Fig. 26. In this 
elevation, none of the faces of the prism are seen in their true size. 
The auxiliary vertical plane, nnjb", could have been revolved about 
that trace, directly hack into the horizontal plane, causing the 
corresponding elevation to appear in the lines D^?, tfcc, produced 
to the left oijb" as a ground line. Elevations on auxiliary vertic.nl 
planes can always be made thus, but it seems more natural to see 
them side by side above the principal ground line, by transferring 
the auxiliary planes as heretofore described. 

50. Fig. 27 represents two elevations of a hexagonal prism, 
placed so as to show the base in one elevation, and three of its 
faces, unequally, in the other. The centre of the elevation which 
shows the base, may be made in a given line perpendicular to o'g\ 
by placing the centre of the circumscribing circle used in con- 
structing the hexagon, upon such a line. Having constructed this 
elevation, project its points, a,6, &c, across to the other vertical 
plane, P', which is in space perpendicular to the plane, P, at the 
line, o'g'. By representing the elevation on P' as touching o ' g\ we 
indicate that the prism touches the plane, P, just as the elevation 
in Fig. 24, indicates that the prism there shown rests upon the 
horizontal plane. 

51. Pbob. 16. — To construct the plan and two elevations of a 
pile of blocks of equal widths, but of different lengths, so placed 
as to form a symmetrical body of uniform width. PI. III., 
Figs. 28, 29. 

Here for example afg is the plan of the lowest step; kbe is that 
of the middle step, and cdh that of the upper step (43). 

The auxiliary vertical plane of projection, perpendicular to the 
horizontal piaffe at tt"f", is made to coincide with the principal 
vertical plane by direct revolution. The point a'" a"", the projec- 



18 PROJECTIONS OF SOLIDS. 

tion of aa' on the auxiliary vertical plane, revolves in a horizontal 
arc, of which a!" a" is the horizontal, and a"" a" the vertical pro- 
jection (31), giving a", a point of the second elevation. Other 
points of this elevation are found in the same way. This figure 
differs from Figs. 18 and 20, of Plate II., only in presenting rrfore 
points to be constructed. If the student finds any difiiculty with 
this example, let him refer to those just mentioned, and to first 
principles. 
x Example. — Construct an elevation on a plane parallel to af 

52. Pbob. 17. — To construct the vertical projection of a vertical 
circle, seen obliquely. PI. HI., Fig. 30. 

Let BF be the given projection of the circle. It is required to 
find its vertical projection, A'B'D'F'. For this purpose, the circle 
must be first brought into a position parallel to a plane of projec- 
tion, since we can then make both of its projections, and hence can 
then take both projections of any point upon it. Let the circle be 
made parallel to the vertical plane. To do this, it only need be 
revolved about any vertical axis. In the figure, the axis is the 
vertical tangent, F — f'F'. After this revolution, the projections 
of the circle are bF — b'c'F'h'. Now taking any point on this circle, 
as aa\ it returns about the axis F — -f'F' in the horizontal arc 
a A — a' A' (31), giving A' by projecting A upon a' A'. Likewise bb r 
returns in the arc bB — b'B to BB'; and cc', which is vertically under 
aa\ returns in the arc cG — c'C to CC. Thus all the points A',B', 
C, etc., being found and joined, we have A'B'C'H', the required 
oblique elevation of the vertical circle FB. 

Examples. — 1st. Let the circle be revolved about its vertical 
diameter HD, or any vertical axis between F and B. 

2d. About any vertical axis ; in the plane BF produced ; or only 
parallel to it. 

3d. Let the circle be made perpendicular to the vertical plane, 
and oblique to the horizontal plane. 

53. Peob. 18. — To construct the projections of a cylinder whose 
convex surface rests on the horizontal plane, and whose axis is in- 
clined to the vertical plane. PI. III., Fig. 31. 

As may be learned from Fig. 19, PI. II., the projection of a right 
cylinder upon any plane to which its axis is parallel, will be a 
rectangle. Therefore let CSTV, PI. ELL, Fig. 31, be the plan of 
the cylinder. Since it rests upon the horizontal plant, ^'w', in the 
ground line, is the vertical projection of its line of contact with that 



PIj.IL 



i9. 



B' 



2D. 




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B' C* D' 



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PROJECTIONS OP SOLIDS. ig 

plane, and jt/A'is the vertical projection ofpA, the highest element 
of the cylinder, as it is at a height above the ground line, equal to 
the diameter, TV, of the cylinder. The vertical projection of either 
base may be found by the last problem. In the figure, the left 
hand base is thus found, and the construction, being fully given, 
needs no further explanation. 

54. The vertical projection of the right hand base TV is found 
somewhat differently. It is revolved about its horizontal diameter, 
TV — T'V, till parallel to the horizontal plane. It will then appear 
as a circle, and a line, as n"n, will show the true height of n above 
the diameter TV. So, also, o"o will show the true distance of o 
below TV. Therefore the vertical projections of the points n and 0, 
will be in the line n — n\ perpendicular to the ground line, and at 
distances above and below T'V', the vertical projection of TV, 
equal, respectively, to nn" and 00". Having, in the same maimer, 
found r' and t', the vertical projections of two points w T hose com- 
mon horizontal projection t — r is assumed, as was n — 0, the vertical 
projection of the base TV can be drawn by the help of the irregu- 
lar curved ruler. 

55. In the execution of this figure, SV is made slightly heavy, and 
TV fully heavy, and the portion, n'T't', of the elevation of the right 
hand base, and the small portion, D V, of the left hand base, are made 
heavy. Suffice it to say : First. That a part of the convex surface 
is in the light, while the right hand base is in the dark. Second. 
n'T't' divides the illuminated half of the convex surface, from the 
base at the right, which is in the dark ; and D V divides the illu- 
minated left hand base from the visible portion of the darkened 
half of the convex surface (18-20). 

-- Example. Let the axis of the cylinder be parallel to the vertical 
plane, only. 

56. Prob. 19. To construct the two projections of a right cone, 
with a circular base in the horizontal plane ; and to construct 
either projection of a line, drawn from the vertex to the circum- 
ference of the base, having the other projection of the same line 
given. PI. in., Fig. 32. 

Remark. When the axis of a cone is vertical, perpendicular to 
the vertical plane, or parallel to the ground line, the cone is shown 
in right projection as much as such a body can be, but as all the 
straight lines upon its surface are then inclined to one or both 
planes of projection, the above problem is inserted here among 
problems of oblique projections. 



20 PROJECTIONS OF SOLIDS. 

Let VB be the radius of the circle, which, with the point V, is 
the horizontal projection of the cone. Since the base of the cone 
rests in the horizontal plane of projection, C'B' is its vertical 
projection. Since the axis of the cone is vertical, V, the vertica 1 
projection of the vertex, must be in a perpendicular to the ground 
line, through V, and may be assumed, unless the height of the 
cone is given. V'C and VB', the extreme elements, as seen ir> 
elevation, are parallel to the vertical plane of projection, hence 
their horizontal projections are CV and BV, parallel to the ground 
line (8 e). Let it be required to find the horizontal projection of 
any element, whose vertical projection, V'D', is given. V is the 
horizontal projection of V, and D', being in the circumference oi 
the base, is horizontally projected at D, therefore YD is the hori- 
zontal projection of that element on the front of the cone, whose 
vertical projection is VD'. V'D' is also the vertical projection of 
an element behind VD, on the back of the cone. Having given, 
YA, the horizontal projection of an element of the cone, let it be 
required to find its vertical projection. V is the vertical projec- 
tion of V, and A, being in the circumference of the base, is verti- 
cally projected at A'. Therefore V'A' is the required vertical pro- 
jection of the proposed line. In inking the figure, no part of the 
plan fe heavy lined, and in the elevation, only the element VB' is 
slightly heavy. 

~~ Examples. — 1st. Construct three projections of a cone placed as 
the cylinder is in Prob. 13. 

2d. As the cylinder is in Prob. 18. 

57. Prob. 20. To construct the projections of a right hexagonal 

prism ; whose axis is oblique to the horizontal plane, and parallel 
to the vertical plane. PI. III., Figs. 33, 34. 

1st. Commence by constructing the projections of the same prism 
as seen when standing vertically, as in Fig. 33. The plan only is 
strictly needed, but the elevation may as well be added here, for 
completeness' sake, and because some use can be made of it. 

2nd, Draw J"G\ making any convenient angle with the ground 
line, and set off upon it spaces equal to G'J', J'H', and JT, from 
Fig. 33. 

3rd. Since the prism is a right one, at J", &c, draw perpen- 
diculars to J^G", make each of them equal to J'C, Fig. 33, and 
draw F'C", which will be parallel to J'G*, and will complete the 
second elevation. 

4th. Let us suppose that the prism was moved from its first 



PROJECTIONS OF SOLIDS. 21 

position, Fig. 33, parallel to the vertical plane, and towards tins 
right, and then inclined, as described, with the corner, CJ', of the 
base, remaining in the horizontal plane. It is clear that all points 
of the new plan, as B'", would be in parallels, as BB'", to the 
ground line, through the primitive plans, as B, of the same points. 
It is equally true that the points of the new plan will be in perpen- 
diculars to the ground line through the new elevations B", &c, of 
the same points (15), hence these points B'", <fcc, will be at th 
intersections of these two groups of lines. Thus, A"' is at the 
intersection of AA'" with A* A"' ; C" is at the intersection of CC"' 
with C"C"; K"' is at the intersection of DK'" with H"K'", &g. 

oth. B'"C", F'"E'", and G'"K"', being the projections of lines 
of the prism which are parallel in space, are themselves parallel. A 
similar remark applies to C'"D", A'"F'", and H'"G'". Observe, 
that as the upper or visible base is viewed obliquely, it is not seen 
in its true size, F"'C"' being less than FC, Fig. 33; so that this 
base A"'C'", E'", does not appear in the new plan as a regular 
hexagon. 

58. Peob. 21. To construct the projections of the prism, given 
in the previous problem, when its edges are inclined to both 
planes of pi'ojection. PI. III., Fig. 34a. 

If the prism, PI. 111., Fig. 34, be moved to any new position, 
such that the inclination of its edges to the vertical plane, only, 
shall be changed, the inclination of its edges to the horizontal 
plane of projection being unchanged, the new plan will be merely 
a copy of the second plan, placed in a new position. Let the par- 
ticular position chosen be such that the axis of the prism shall be 
in a plane perpendicular to the ground line, i.e. to both planes of 
projection ; then the axis of symmetry, C'"G'", of the second 
plan, will take the position C""G'"', and on each side of this line the 
plan, Fig. 34 a, will be made, similar to the halves of the plan in 
Fig. 34. 

As the prism is turned horizontally about the corner J", and 
then transferred, producing the result that the inclination of its 
axis to the horizontal plane is unchanged, all points of the third 
elevation, as A'"", C""', <fcc., will be in parallels to the ground line 
through A*, C", &c., and in perpendiculars to the ground line, 
through A"", C"", Ac. 

By examination of this solution, and by inspection of Figs. 34 
and 34a, it appears that a change in the position of the axis, 
with reference to but one plane of projection at a time, can be 



22 PROJECTION'S OF SOLIDS. 

represented directly from projections already given; aLo that a 
carve, beginning with the first plan, and traced through the six 
figures composing the three given pairs of projections in the order 
in which they must be made, would be an S curve, ending in the 
third elevation. 

59. Execution. — The full explanation of the location of the heavy 
lines cannot here be given. The careful inquirer may be able to 
satisfy himself that the heavy lines of the figures, as shown, are the 
projections of those edges of the prism which divide its illuminated 
from its dark surfaces. 

60. Prob. 22. To construct the projections of a regular hexa- 
gonal pyramid, whose axis is inclined to the horizontal plane only. 
PI. m., Figs. 35, 36. 

1st. Commence, as with the prism in the last problem, by repre- 
senting the pyramid as having its axis vertical. 

2nd. Draw a"d", equal to a'd', and divided in the same way. At 
n", the middle point ofa"d", draw n"Y" perpendicular to a"d", 
and make it equal to n'Y', which gives V" the new elevation of the 
vertex. Join V" with a", b", c", and d" , and the new elevation 
will be completed. 

3rd. Supposing the same translation and rotation to occur to the 
primitive position of the pyramid, that was made in the case of the 
prism (57, 4£A), the points of the new plan, Fig. 36, will be found 
in a manner similar to that shown in Fig. 34. V" is at the inter- 
section of VY'"with V"V"; c'" is at the intersection cc" with 
c"c" \ d'" is at the intersection ofdd'" with d"d'", &c. 

4tth. The points, d"b'"c" .... f"\ of the base, are connected 
with V", the new horizontal projection of the vertex, to complete 
the new plan. If the pyramid were less inclined, the perpendicular 
V"V" would fall within the base, and the whole base would then 
be visible in the plan. As it is, f'"d" and a"'b'" are hidden, and 
therefore dotted. 

5th. The heavy lines are correctly placed in the diagram ; also 
the partially heavy lines, which are all between Y'"d'" and the 
ground line, but the reasons for their location cannot here be given, 
beyond the general principle (18-20) already given. 

61. Peob. 23. To construct the projections of the regular hexa- 
gonal pyramid, when its axis is oblique to both planes of projeo 
tion. PI. HI., Fig. 36a. 

Suppose the pyramid here shown to be the one represented in 



ELEMENTARY INTERSECTIONS. 28 

figures 35 and 36, and suppose that it has been turned about any 
vertical line as an axis. Then, first, every point of it will move 
horizontally; second, every point will hence remain at the same height 
as before ; third, therefore, the inclination of all the edges to the 
horizontal plane will be unchanged ; and hence, fourth, the new 
plan, Fig. 36a, will be only a copy of the second plan, Fig. 36, 
placed so that its axis of symmetry, N""d"", shall make any as- 
sumed angle with the ground line. 

By {second) and (15) the points, as V"", of the third elevation, 
will be at the intersection of parallels to the ground line, through 
the corresponding points, as V", of the second elevation, with per- 
pendiculars through the same points, as Y //// , seen in the third plan. 
Observe that the two points vertically projected in c", being at the 
same height above the ground line, will appear in the third eleva- 
tion at </"" and e"'", in the same straight line, through c", and par- 
allel to the ground line. (32). 

Remembering also that lines which are parallel in space must 
have - parallel projections, on the same plane, c!""d'"" will be paral- 
lel X>of'""a'"", &c. The heavy lines are indicated in the figure. 

Example. — Construct Fig. 36a from Fig. 36, without a new plan, 
by taking a new vertical plane with its ground line parallel to 
Y""d"", and revolving it directly back as mentioned in (49). 

§ IY. — Special Elementary Intersections and Developments. 

62. The positions of other planes, than those of projection, are 
indicated by their intersections with the planes of projection. 
These intersections are called traces. 

A plane can cut a straight line in only one point ; hence, if a 
plane cuts the ground line at a certain point, its traces, both being 
in the plane, must meet in that point. 

In PI. I., Fig. 5, bBb'k' is a plane perpendicular to the ground 
line, MG, and, therefore, to both planes of projection, and we see 
that its two traces, bk f and b'k f , are perpendicular to the ground 
line at Jc'. Likewise in PI. I., Fig. 15, Aaa't is a plane perpendicu- 
lar to the ground line MQ, and its traces at and at are perpendi- 
cular to MQ. That is: if a plane is perpendicular to the ground 
line, its traces will also be perpendicular to that line. 

This is seen in regular projection, in PI. I., Fig. 10, where PQ is 
the horizontal trace, and P'Q, the vertical trace, of such a plane. 

In PI. I., Fig. 5, FKfk is a plane, parallel to the vertical plane, 
and it has only a horizontal trace, fh, which is parallel to the 
ground line. The same is true for all such planes. Likewise, 



SJ4 ELEMENTARY INTERSECTIONS. 

ABa'b' is a horizontal plane. All such planes have only a vertical 
trace, as a'b', parallel to the ground line. 

In PI. I., Fig. 2, the plane Yv'd'd is perpendicular only to the 
vertical plane, and, as the figure shows, the horizontal trace only, 
as dd', of such a plane, is perpendicular to the ground line. Also 
the angle v'd'b', between the vertical trace, v d', and the ground 
line, is the angle made by the plane with the horizontal plane. 

In like manner, it can easily be seen that, if a plane be perpen 
dicular only to the horizontal plane, as in case of a partly open door, 
its vertical trace only (the edge of the door at the hinges) tcill be 
perpendicular to the ground line, and the angle between its hori- 
zontal trace and the ground line, will be the angle made by the 
plane with the vertical plane of projection. 

Finally, if a plane is oblique to both planes of projection, both 
of its traces will be oblique to the ground line, and at the same 
point. Thus, PL I., Fig. 6, may represent such a plane, having LF 
for its horizontal, and L'F for its vertical trace. 

All the principles just stated can be simply illustrated by taking 
a book, half open, for the planes of projection, and either of the 
triangles for the given movable plane ; and when clearly under- 
stood, the following problems can also be easily comprehended. 

Prob. 24. — To find the curve of intersection of a cylinder with 
a plane. PI. IV., Tig. 1. 

Let the cylinder, ADBG — A'B", be vertical, and the cutting 
plane, PQP', be perpendicular only to the vertical plane. All 
points in such a plane must have their vertical projections (that is, 
must be vertically projected) in the vertical trace, QP', of -the 
plane, but the required curve must also be embraced by the visible 
limits, A'A" and B'B", of the cylinder. Hence, a'b' is the verti- 
cal projection of this curve. Again, as the cylinder is vertical, all 
points on its convex surface must be horizontally projected in 
ADBG. Hence, this circle is the horizontal projection of the 
required curve. 

Prob. 25. — To revolve the curve found in the last problem, so as 
to show its true size. 

When a plane revolves about any line in it as an axis, every 
point of it, not in the axis, moves in a circular arc, whose radii are 
all perpendicular to the axis. The representation of the revolution 
is much simplified by taking the axis in, parallel to, or perpendicu 
far to, a plane of projection (31). 



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ELEMENTARY INTERSECTIONS. 25 

Let AB — a'b', the longer axis of the curve, and which is parallel 
to the vertical plane of projection, be taken as the axis of revolu- 
tion. The curve may then be revolved till parallel to that plane, 
when its real size and form will appear. Then, at c\ d\ <fcc, the 
vertical projections of C and H, D and G, etc., draw perpendicu- 
lars, as c"h", to a'b\ and make e'e" = c'h" —nC Proceed likewise 
at d\ &c, since the lines, as nC, being parallel to the horizontal 
plane, are seen in their true size in horizontal projection ; and join 
the points a'h"g", tvjc, which will give the required true form and 
size of the curve of intersection before found. 

Example. — This curve is an oval, called an ellipse. Its true size 
could have been shown by revolving its original position about 
DG as an axis, till parallel to the horizontal plane. The student 
may add this construction to the plate. 

Prob. 26. — To develqpe the portion of the cylinder, PI. IV., 
Fig. 1, below the cutting plane, PQP'. 

The convex surface of a cylinder is wholly composed of straight 
lines, called elements, parallel to its axis. The convex surface of a 
cone is composed of similar elements, all of which meet at its vor- 
tex. Hence, each of these surfaces can evidently be rolled upon a 
plane, till the element first placed in contact with the plane, returns 
into it again. The figure, thus rolled over on the plane, is called 
the development of the given convex surface, and its area equals 
the area of that surface. 

Suppose the cylinder to be hollow as if made of tin, and to be 
cut open along the element B 6'. Then suppose the element AV 
to be placed on the paper, as at A' a', Fig. 2, and let each half be 
rolled out upon the paper. The part ADB will appear to the left 
of A' a', and the part AGB, to the right. The base being a circle, 
perpendicular to the elements, will develope into a straight line 
B'B", Fig. 2, found by making A'c = AC, Fig. 1, cd=GD, Fig. 1, 
<fcc, and A'A=AH, Fig. 1, &c. B'B" may also, for convenience, 
be A'B', Fig. 1, produced. Then the parallels to AV, through c 
d, <fcc, will be developments of elements standing on C, D, &c, 
Fig. 1, and by projecting over upon them, a' at a\ c' at c' and h' ; 

d' at d' and g\ B' at b' and b'\ and joining the points, tht 

figure 'B'Wb"a'b', will be the required development of the cylinder- 

Remark. — If, now, a flat sheet of metal be cut to the patterr 
just found, it will roll up into a cylinder, cut off obliquely as by 
the plane PQP'. By making the angle P'QA' of any desired size, 
the corresponding flat pattern can be made as now explained. 



26 ELEMENTARY INTERSECTIONS. 

Peob. 27. — To find the intersection of a vertical cone, with a 
plane, perpendicular to the vertical plane of projection. PI. IV., 
Fig. 3. 

Let V — ADBC be the plan, and A'B'V the elevation of the 
cone, and PQ and P'Q the traces of the given cutting plane; whose 
horizontal trace, PQ, shows it (62) to be perpendicular, to the ver- 
tical plane. For the reasons given in Problem 25, a'b' will be the 
vertical projection of the required curve. The convex surface of 
the cone not being vertical, the horizontal projection of the inter- 
section will be a curve, which must be found by constructing its 
points as follows. 

First. The method by elements. Any line, as VE', is the verti- 
cal projection of two elements whose horizontal projections are 
VE and VF (Prob. 19). Therefore e', where it crosses the vertical 
projection, a'b', of the intersection, is the vertical projection of two 
points of the required curve. Their horizontal prp-ections, e and/, 
are found by projecting e' down upon VE and VF. Other points 
can be found in the same manner, except d and g, since the pro- 
jecting line d'd coincides with the elements VD and VG. The 
horizontal projections of a' and b' are a and b. 

Seco?id. The method by circular sections. Let M'N' be the ver- 
tical trace of a horizontal auxiliary plane through d'. This plaue 
will cut from the cone the circle m'n' — dmg, on which d' can be 
projected at d and g, the points desired. Other points of the 
horizontal projection can be found in the same manner. 

Remarks. — a. The curve adbg — a'b' is an ellipse whose longer 
axis is the line ab — a'b'; whose true length is a'b'. Its shorter 
axis is the line pq — p', whose true length pq bisects ab, and is 
always less than ab; since it is a chord of the circle x'y' through 
p', and x'y' is easily seen to be equal to ab. An ellipse, having 
thus two axes of symmetry, can be drawn by using an arc of the 
irregular curve that will fit one quarter of it. 

b. On the cylinder, d', the middle of a'b' is on the axis O — dd'. 
That is, the centre of the ellipse cut from a cylinder, is on the axis 
of the cylinder. Not so, however, with the cone; p', the middle 
of a'b', is not on V'D', the vertical projection of the axis, but is on 
the side of it towards the lowest point, bb', of the curve of inter- 
section. On account of the acuteness of the intersections at p and 
q, these points can better be found as were d and g. 

Examples. — 1st. To make the horizontal projection less circular 
than in the figure, let the cone be quite flat, as at AVB, Fig. 9, and 
with a! near the vertex, and b' quite near the base. 



ELEMENTARY INTERSECT JONS. 27 

2d. Find the true size of the curve by either of the ways indi- 
cated in Prob. 25, also by revolving the plane PQP', containing 
it, either, about PQ as an axis, into the horizontal plane; or, 
about P'Q as an axis, into the vertical plane. In the former case 
it is only to be remembered that e'Q, for example, shows the tru<= 
distance of ee' from PQ; and, in the latter case, that ek, for exam 
pie, shows the true distance of ee' from the vertical trace P'Q (6), 

Prob. 28. — To develope the convex surface of a cone, PL IV., 
Fig. 4, together with the curve of intersection, found in the last 
problem. 

First. If the element VU — V'B' be placed in contact with the 
paper at V'B', and if the cone be then rolled upon the paper till 
this element returns into it again, as at V'B", the development, 
V'B'B", will be made. As all the elements are equal, and hs the 
vertex is stationary, the development of the base will be the arc 
B'B", with a radius equal to V'B', the cone's slant height and of a 
length equal to the circumference ADBG. This length is found, 
as in case of the cylinder, by taking equal arcs of the base, so 
small that their chords shall be sensibly equal to them, and laying 
off those chords from B', on the arc B'B", till B" is located. 
Thus, BE being one eighth of ADBG, its length is laid off as at 
BV eight times to find B". 

Second. To show the curve, adbg — a'b', on the development, 
consider that only the extreme elements, as VB — V'B', show their 
true length in projection. Hence, the points between a' and b' 
must be revolved around the axis of the cone, into these elements, 
in order to show their true distances from the vertex. This axis 
being vertical, the arcs of revolution will be horizontal, and will 
therefore be vertically projected in the horizontal lines c'u, d'n, &c, 
and Yu, V/i, &c, will be the true distances of c', d', &c, from the 
vertex. Hence, make Y'a" — Y'a'; Y'u', and Y , u' / =V'u; W, and 
Y'n" =Yn, &c, and the curve b'a"b'' will be the development of 
the intersection of the plane PQP' with the cone. 

JRemark. — The remarks made upon the development of the 
cylinder equally apply here. 

Prob. 29. — To find the intersection of a vertical cylinder with 
two horizcnital ones ; their axes being in a plane parallel to the ver- 
tical plane of projection. PI. IV., Fig. 5. 

ABE— A'B'A'B" is the vertical cylinder, and MNQR— O'P'O 1 
P* the lower horizontal cylinder. 



28 ELEMENTARY INTERSECTIONS. 

First. To find the highest and lowest, and foremust and hind 
most points of the intersection. Since the horizontal cylinder if 
the smaller one, it will enter the vertical cylinder on one side, aid 
leave it on the other, giving two curves; but as one cylinder is ver- 
tical, and the intersection, being common to both, is on it, the 
horizontal projections of both curves are known at once to be 
CAE and DBF. Now A is the horizontal projection of both the 
highest and lowest points of CAE. Their vertical projections are 
a" and a'. Also C and E are the horizontal projections of the fore- 
most and hindmost points, and c\ on M'N', midway between O 
and O", is the vertical projection of both of them. (41.) 

In like manner b\ b" and d' are found. 

Second. To find other intermediate points. Take the two points 
whose horizontal projection is G, for example. They are on the 
horizontal elements, one on the upper, and the other on the lower 
half of the horizontal cylinder, and whose horizontal projection is 
ST. But to find their vertical projections, we must revolve one oi 
the bases, as MQ, till parallel to a plane of projection. Let this 
base revolve about its vertical diameter, O — O'O", till parallel to 
the vertical plane, when OM" — 0'M'"0" will be the vertical pro- 
jection of its front half. In this revolution the points, S, revolve 
to S", and will thence be vertically projected at TJ' and S'. In 
counter revolution, these points return in horizontal arcs to u' and 
«', and u'v' and s't' are the vertical projections of the elements ST. 
Hence, project G, and also H, at g' and g", h' and h\ and we shall 
have four more points of intersection. Any number of points can 
be similarly found. 

Examples. — 1st. The last four points could as easily have been 
found by revolving the base MQ about the horizontal diameter 
MQ — M', till parallel to the horizontal plane. This construction 
is left for the student. 

2c?. If the axes did not intersect each other, as at IF, the points 
C and E would not be equidistant from OP, and would not have 
one point, c', for their vertical projection, and the vertical projec- 
tion of the back half, as AE, of each curve would be a dotted line, 
separate from the same projection of the front half. The student 
may construct this case, also that where one of the elements, MN, 
or QR, does not intersect the vertical cylinder. 

3d. The horizontal cylinder, Fig. 6, shows that when the two 
cylinders, placed as before, are of equal diameter, the vertical pro- 
jections of their curves of intersection are straight lines. Hence, 
each of the curves themselves is contained in a plane, that is, it is 



ELEMENTARY INTERSECTIONS. 



29 



a "plane curve." This figure, if regarded separately, as a plan 
view, therefore may represent the plan of the intersection of two 
equal semi-circular arches, and the curves, KL and AY, of inter- 
section, will be ellipses. 

The curves on the cylinders in Fig. 5 cannot be contained in 
planes. Such curves are said to be of double curvature. 

4th. By developing the cylinders, in Figs. 4 and 5, as in Fig. 2, 
the patterns may be found which will give intersecting sheet metal 
pipes, when rolled up in cylindrical form. The student should 
construct these developments, also the case in which the vertical 
cylinder should be the smaller one. 



Prob. 30. — To find the intersection of a horizontal cylinder with 
a vertical cone. PI. IV. , Fig. 7. 

Let ABV be the vertical projection of a cone, and let the circle 
with radius o«, be an end view of the cylinder ; its axis, o'o", in- 
tersecting A'V', that of the cone. Let PQ be the vertical trace 
of a second vertical plane, perpendicular to the ground line, as 
in PI. I., Figs. 15 and 16, and let V'E'D'be the vertical projection 
of the cone, and Gr'Gr"N'N" that of the cylinder, on this plane. 
In this construction, therefore, two vertical projections are em- 
ployed, instead of a horizontal and vertical projection, for any two 
projections of an object are enough to show its form and position. 
This will more readily appear by turning the plate to bring VAB 
below PQ, when PQ will be the ground line, the right hand pro- 
jection a plan, and the left hand one an elevation, like Fig. 5. 

Now to find the intersection. Speaking as if facing the vertical 
plane of projection, represented by the paper to the left of PQ, 
after revolving that plane about PQ into the paper, AY — A'V' is 
the foremost element, and a' is found by projecting a across upon 
A'V'. Next, DV is the right hand projection of two elements, 
whose left hand projections are E'V and D'V. We therefore 
project G at// and e', 

To find intermediate points. Assume any element FV, draw 
FF" perpendicular to AB, then make an arc, AF", of the plan or' 
the cone's base, and make A'F' = A'H' — FF". Then V'F' and 
V'H' will be the left hand projections of the two elements project- 
ed in FV. Then project/ at /'and h' on these elements, and^VA V 
will be the visible part of the intersection. Its right hand projec- 
tion is a/Gr, where /and G are, each, the projection of two point* 
on opposite sides of the cone. 



30 ELEMENT ARY INTERSECTIONS. 

Example. — By developing the cone and the cylinder, patterns 
could be made for a conical pipe entering a cylindrical one. 

63. Observing that, in every case, the auxiliary planes are 
made to cut the given curved surfaces in the simplest manner, that 
is, in straight lines or circles, we have the following principles. To 
cut right lines, at once, from two cylinders, as in Fig. 5, a plane 
must be parallel to both their axes. To cut a cylinder and cone, at 
once, in the same manner, as in Fig. 7, each plane must contain the 
vertex of the cone, and be parallel to the axis of the cylinder. To 
cut elements at once from two cones, a plane must simply contain 
both vertices. 

Examples. — 1st. Thus, in Fig. 10, all planes cutting elements, both 
from cone V V, and cone AA', will contain the line VAB, hence 
their traces on the horizontal plane will merely pass through B. 
Thus the plane BD cuts from the cone, V, the elements W — Ya, 
and W — Yc; and from the cone, A, the elements A'D' — Ad, and 
&!d' — AD. The student can complete the solution, the remainder 
of which is very similar to the two preceding. 

2d. To find the intersection of a sphere .and cone, PI. IV., 
Fig. 1 1 , auxiliary planes may most conveniently be placed in two ways. 
First, horizontally. Then each will cut a circle from the sphere, 
and one from the cone ; whose horizontal projections will be circles, 
and whose intersections will be points of the intersection of the 
cone and sphere. Second, vertically. Then each plane must con- 
tain the axis of the cone, from which it will cut two elements. It 
will also cut the sphere in a circle, and by revolving this plane 
about the axis of the cone till parallel to the vertical plane, as in 
Prob. IV, the intersection of the circle with the revolved elements, 
see Prob. 27, may be noted, and then revolved back to their true 
position. The student can readily make the construction, after due 
familiarity with preceding problems has made the apprehension of 
the present article easy. 

Prob. 31. — To find the intersection of a vertical hexagonal 
prism with a sphere, whose centre is in the axis of the prism. PI. 
IV., Fig. 8. 

Let O — ABC be part of the sphere, and DGHK the prism, 
showing one face in its real size, and therefore requiring no plan 
(47). Draw dg parallel to AC, and the arc ehfwith O as a centre, 
and through e and/. This arc is the real size of the intersection 
of the middle face of the prism with the surface of the sphere. All 
the faces, beins: equal, have circular tops, equal to ehf; but, being 



ELEMENTARY INTERSECTIONS. 31 

seen obliquely, they would be really elliptical in projection. It is 
ordinarily sufficient, however, to represent them by circular arcs, 
tangent to hn, the horizontal tangent at f h and containing the 
points d and e, and f and g, as shown. 

Remark. — The heavy lines here, show the p.*rt of the prism 
within the sphere, as a spherical topped bolt hcs-vL To make 
Dc?=EF, draw Od at 45° with AC, to locate dg. To make the 
spherical top flatter, for the same base DG, take a inrger sphere, 
and a plane above its centre for the base of the prism. 

Pkob. 32. — To construct the intersection of a vertical cone with 
a vertical hexagonal prism y both having the same axis. PL IV., 
Fig. 9. 

Lejb YAB be the cone, and CFGH, the prism, whose elevation 
can be made without a plan (48), since one face is seen in its real 
size. The semicircle on cf is evidently equal to that of the cir- 
cumscribing circle of the base of the prism, and ct is the chord of 
two thirds of it. Then half of ct, laid off on either side of O, the 
middle of CF, as at Ow, will give np.the projection of the middle 
face EDc? after turning the prism 90° about its axis. This done, 
np will be the height, abo've the base, of the highest point at which 
this and all the faces will cut the cone. A vertical plane, not 
through the vertex of a cone, cuts it in the curve, or " conic sec- 
tion," called a hyperbola. The vertical edges of the prism cut the 
cone at the height Ff hence, drawing the curves, as dse, sharply 
curved as at 5, and nearly straight near d and e, we shah have a 
sufficiently exact construction of the required intersection. 

JRemark. — The heavy lines represent the part of the prism within 
the cone, finished as a hexagonal head to an iron "bolt," such as ia 
often seen in machinery. The horizontal top, hg, of the head, may 
be drawn by bisecting pr at g. To make Cc=ED, as is usual in 
practice, simply draw Oc at an angle of 45° with AB, to locate cf. 
By making VAB = 30° perhaps the best proportions will be found. 

64.. In the subsequent applications of projections in practical 
problems, the ground line is very generally omitted; since a know- 
ledge of the object represented makes it evident, on inspection, 
which are the plans, and which the elevations. 

General Examples. 
The careful study of the detailed explanations of the preced- 
ing problems, will enable the student to solve any of the follow- 
ing additional examples. 



32 ELEMENTARY INTERSECTIONS. 

Ex. 1. — In Prob. 24, substitute for the cylinder any prism, 
find the intersection with the plane PQP', and, by Prob. 25, find 
the true form and size of this intersection. 

Ex. 2. — In Prob. 27, substitute for the cone any pyramid. 
Vary this and Ex. 1 by different positions of PQP', cutting 
both bases in Ex. 1. 

Ex. 3.— In Ex. 2, find, by Prob. 25 or by Prob. 27, Ex. 2d, 
the true form and size of the intersection and, by Prob. 28, the 
development of the convex surface of the pyramid. 

Ex. 4. — In Probs. 22, 23, substitute for the pyramid a cone 
whose convex surface, rolling on H (23), shall be shown, first, 
with its axis parallel to V; and, second, with its axis oblique to V. 

Ex. 5. — In Ex. 4, find the intersection of the cone with any 
plane parallel to H; and show the curve on both positions of the 
cone. 

Ex. 6. — In Ex. 5, let the cutting plane be vertical but ob- 
lique to V, and not containing the cone's vertex. 

Ex. 7. — In Prob. 29, let the horizontal cylinder be the large: 
one, and, after finding its intersection with the vertical one, de- 
velope it. 

Ex. 8. In Probs. 22, 23, substitute for the pyramid a cylinder. 

Ex. 9. — In Probs. 22, 23, substitute for the pyramid a hollow 
hemisphere. 

Ex. 10. — In Prob. 29, let the axis of the horizontal cylinder be 
inclined first to H only, and then to both H and V. 

Ex. 11.— In Probs. 22, 23, let the pyramid, when in the posi- 
tion shown in Fig. 36 (but more inclined), rest its edge V'V" 
against, an upper edge of a cube standing on H. 

Ex. 12. — Find the four following sections of a sphere: one by 
a horizontal plane, one by a plane parallel to V, one by a vertical 
plane oblique to V, and one by a plane perpendicular to V and 
oblique to H. 

Ex. 13. — Out a regular hexagon from a cube. 

Ex. 14.— Out a rhombus and an isosceles triangle from the 
square prism. PI. II., Fig. 17. 

Ex. 15.— Construct the projections of the cylinder, PI. IV., 
Fig. 1, after rotating it and PQP', together, 45° on its axis. 

Ex. 16.— Substitute for the blocks, PI. III., Figs. 28, 29, a 
pile of thin cylinders of unequal diameters, but with a common 
axis placed obliquely to V. 



PL. IV 




DIVISION SECOND. 

DETAILS OF MASONRY, TTOOD, AND METAL CONSTRUCTIONS. 



CHAPTER I. 

CONSTRUCTIONS IN MASONRY. 

§ I. — General Definitions and Principles applicable both to Brick 
and Stone-work. 

65. A horizontal layer of brick, or stone, is called a course. The 
seam between two courses is called a coursing-joint. The seam 
between two stones or bricks of the same course, is a vertical or 
heading-joint. The vertical joints in any course should abut against 
the solid stone or brick of the next courses above and below. This 
arrangement is called breaking joints. The particular arrangement 
of the pieces in a wall is called its bond. As far as possible, stones 
and bricks should be laid with their broadest surfaces horizontal. 
Bricks or stones, whose length is in the direction of the length of a 
wall, are called stretchers. Those whose length is in the direction 
of the thickness of a wall, are called headers. 

§ II.— Brick Work. 

66. If it is remembered that bricks used in building have, usually, 
nn invariable size, 8" x 4" x 2" (the accents indicate inches), and 
that in all ordinary eases they are used whole, it will be seen that 
brick walls can only be of certain thicknesses, while, in the use of 
stone, the wall can be made of any thickness. 

Thus, to begin with the thinnest house wall which ever occurs, 
viz. one whose thickness equals the length of a brick, or 8 inches ; 
the next size, disregarding for the present the thickness of mortar 
would be the length of a brick added to the width of one, or equal 
to the width of three bricks, making 12 inches, a thickness employed 
in the partition walls and upper stories of first class houses, or the 



CONSTRUCTIONS IN MASONRY. 

outside walls of small houses. Then, a wall whose thickness is 
equal to the length of two bricks or the width of four, making 1 6 
inches, a thickness proper for the outside walls of the lower stories 
of first class houses ; and lastly, a wall whose thickness equals the 
length of two bricks added to the width of one ; or, equals the 
width of five bricks, or 20 inches, a thickness proper for the base- 
ment walls of first class houses, for the lower stories of few-storied, 
heavy manufactory buildings, &g. 

6V. In the common bond, generally used in this country, it may 
be observed — 

a. That in heavy buildings a common rule appears to be, to have 
one row of headers in every six or eight rows of bricks or courses, 
Le. five or seven rows of stretchers between each two successive 
rows of headers ; and, 

b. That in the 12 and 20 inch walls there may conveniently be a 
row of headers in the back of the wall, intermediate between the 
rows of headers in the face of the wall, while in the 8 inch and 16 
inch walls, the single row of headers in the former case, and the 
double row of headers in the latter, would take up the whole thick- 
ness of the wall, and there might be no intermediate rows of 
headers. 

c. The separate rows, making up the thickness of the wall in an} 
one layer of stretchers, are made to break joints in a horizontal 
direction, by inserting in every second row a half brick at the end 
of the wall. 

68. Calling the preceding arrangements common bonds, let us 
next consider the bonds used in the strongest engineering works 
which are executed in brick. These are the English bond and the 
Flemish bond. 

The English Bond. — In this form of bond, every second course, 
as seen in the face of the wall, is composed wholly of headers, the 
intermediate courses being composed entirely of stretchers. Hence, 
in any practical case, we have given the thickness of the wall and 
the arrangement of the bricks in the front row of each course, and 
are required to fill out the thickness of the wall to the best advantage. 

The Flemish Bond. — In this bond, each single course consists 
of alternate headers and stretchers. The centre of a header, in 
any course, is over the centre of a stretcher in the course next 
above or below. The face of the wall being thus designed, it 
remains, as before, to fill out its thickness suitably. 

69. Example 1. To represent an Eight Inch Wall in Eng- 
lish Bond. Let each course of stretchers consist of two rows, sida 

3 



CONSTRUCTIONS IN MASONRY. 35 

by side, the bricks in which, break joints with each other hori- 
zontally. Then the joints in the courses of headers, will be distant 
half the width of a brick from the vertical joints in the adjacent 
courses of stretchers, as may be at once seen on constructing a 
diagram. 

70. Ex. 2. To represent a Twelve Inch Wall in English 
Bond. See PL V., Fig. 37. In the elevation, four courses are 
ehown. The upper plan represents the topmost course, and in the 
lower plan, the second course from the top is shown. The courses 
having stretchers in the face of the wall, could not be filled out by 
two additional rows of stretchers, as such an arrangement would 
cause an unbroken joint along the line, «5, throughout the whole 
height of the wall — since the courses having headers in the face, 
must be filled out with a single row of stretchers, in order to make 
a twelve inch wall, as shown in the lower plan. 

In order to allow the headers of any course to break joints with 
the stretchers of the same course, the row of headers may be filled 
out by a brick, and a half brick — split lengthwise — as in the upper 
plan ; or by two three-quarters of bricks, as seen in the lower 
plan. 

71. Ex. 3. To represent a Sixteen Inch Wall in English 
Bond. The simplest plan, in which the joints would overlap pro- 
perly, seems to be, to have every second course composed entirely of 
headers, breaking joints horizontally, and to have the intermediate 
courses composed of a single row of stretchers in the front and 
back, with a row of headers in the middle, which would break 
joints with the headers of the first named courses. If the stretcher 
courses were composed of nothing but stretchers, there would 
evidently be an unbroken joint in the middle of the wall extending 
through its whole height. 

72. Ex. 4. To represent an Eight Inch Wall in Flemish 
Bond. PI. V., Fig. 38, shows an elevation of four courses, and the 
plans of two consecutive courses. The general arrangement of both 
courses is the same, only a brick, as AA', in one of them, is set six 
inches to one side of the corresponding brick, B, of the next course 
— measuring from centre to centre. 

73. Ex. 5. To represent a Twelve Inch Wall in Flemish 
Bond. PL V., Fig. 39, is arranged in general like the preceding 
figures, with an elevation, and two plans. One course being arranged 
as indicated by the lower plan, the next course may be made up in 
two ways, as shown in the upper plan, where the grouping shown 
at the right, obviates the use of half bricks in every second course. 



36 CONSTRUCTIONS IN MASONRY. 

There seems to be no other simple way of combining the bricks in 
this wall so as to avoid the use of half bricks, without leaving opeD 
spaces in some parts of the courses. 

74. Ex. 6. To represent a Sixteen Inch Wall in Flemish 
Bond. PL V., Fig. 40. The figure explains itself sufficiently. 
Bricks may not only be split crosswise and lengthwise, but even 
thicknesswise, or so as to give a piece 8x4x1 inches in size. 
Although, as has been remarked, whole bricks of the usual dimen- 
sions can only form walls of certain sizes, yet, by inserting frag- 
ments, of proper sizes, any length of wall, as between windows and 
doors, or width of pilasters or panels, may be, and often is, con- 
structed. By a similar artifice, and also by a skilful disposition of 
the mortar in the vertical joints, tapering structures, as tall chim- 
neys, are formed. 

§ HI.— Stone Work. 

75. The following examples will exhibit the leading varieties of 
arrangement of stones in walls. 

Example 1. Regular Bond in Dressed Stone. PL VI. , Fig. 41. 
Here the stones are laid in regular courses, and so that the middle 
of a stone in one course, abuts against a vertical joint in the course 
above and the course below. In the present example, those stones 
whose ends appear in the front face of the wall, seen in elevation, 
take up the whole thickness of the wall as seen in plan. 

The right hand end of the wall is represented as broken down in 
all the figures of this plate. Broken stone is represented by a 
smooth broken line, and the under edge of the outhanging part of 
any stone, as at n, is made heavy. 

76. Ex. 2. Irregular Rectangular Bond. PL VI., Fig. 42. In 
this example, each stone has a rectangular face in the front of the 
wall. These faces are, however, rectangles of various sizes and 
proportions, but arranged with their longest edges horizontal, and 
also so as to break joints. 

77. That horizontal line of the plan which is nearest to the lower 
border of the plate, is evidently the plan of the top line of the ele- 
vation, hence all the extremities, as a\ b', <fcc, of vertical joints, found 
on that line, must be horizontally projected as at a and b, in the 
horizontal projection of the same line. 

78. Ex. 3. Rubble Walls. The remaining figures of PL VI., 
represent various forms of " rubble " wall. Fig. 43 represents a 
wall of broken boulders, or loose stones of all sizes, such as are 
found abundantly in New England. Since, of course, such stones 



















PLV 


- 




_L _J 






1 






1 


B' 










~\ 


i 1 




1 A'| 






' 




B 






















\ 






A 










HI 




1 1 1 














r 


1 


1 




1 1 



1 1 1 1 1 \ 


ii i i i i ii i i ; 


i i i i i / 


i i i i i i i i i i 1 1 


























a b 

























II II II 1 \ 


1 II II 11/ 


II II II if 


II II II 11/ 
















! 




io- 




\ 






I 




















1 











n 



TEL 



TTL 

i r 



IZI 



J_L 



J L 



TTL 



TTL 



I r 



LTL 

TTL 



TTL 
JTTL 



i r 



^5 



TTL 



TTL 



i r\ 



CONSTRUCTION'S IN MASONRY. 37 

would not fit together exactly, the "chinks" between them are 
filled with small fragments, as shown in the figure. Still smaller 
irregularities in the joints, which are not thus filled, are repre- 
sented after tinting by heavy strokes in inking. Fig. 44 repre- 
sents the plan and elevation of a rubble wall made of slate ; 
hence, in the plan, the stones appear broad, and in the elevation, 
long and thin, with chink stones of similar shape. Fig. 45 
represents a rubble wall, built in regular courses, which gives 
a pleasing effect, particularly if the wall have cut stone corners, 
of equal thickness with the rubble courses. 

Ex. 4. A Stone Box-culvert. PI. VI., Figs. 0, D, 
E. Scale T 3 g- of an inch to 1 ft. 

Fig. is a longitudinal section; D, part of an end elevation; 
and E, part of a transverse section. Waste water flowing over 
the dam del', into the well a between the wing-walls a and b' and 
the head li, escapes by the culvert cc — c" , which is strengthened 
by an intermediate cross- wall m" , occurring in the course of its 
length. 

The masonry rests on a flooring of 2-inch planks lying trans- 
versely on longitudinal sills, which, in turn, rest on transverse sills. 
Thus a firm continuous bearing is formed which prevents un- 
equal settling of the masonry, while washing out underneath is 
provided against by sheet piling partly shown at p, p', p" , and 
extending six feet into the ground. 

The student should construct this example on a larger scale, 
from 4 to 6 sixteenths of an inch to a foot; and should add a 
plan, or a horizontal section, both of which may easily be con- 
structed from the data afforded by the given figures. 

79. Execution. — Plate VI. may be, 1st, pencilled; 2d, inked 
in fine lines; 3d, tinted. The rubble walls, having coarser lines 
for the joints, may better be tinted, before lining the joints in 
ink. 

Also, in case of the rubble walls, sudden heavy strokes may be 
made occasionally in the joints, to indicate slight irregularities 
in their thickness, as has already been mentioned. 

The right hand and lower side of any stone, not joining an- 
other stone on those sides, is inked heavy, in elevation, and on 
the plans as usual. The left-hand lines of Figs. 43 and 44 are 



38 CONSTRUCTIONS IX MASONRY. 

tangent at various points to a vertical straight line, walls, such as 
are represented in those figures, being made vertical, at the fin- 
ished end, by a plumb line, against which the stones rest. 

The shaded elevations on PI. VI. may serve as guides to the 
depth of color to be used in tinting stone work. The tint actu- 
ally to be used, should be very light, and should consist of gray, 
or a mixture of black and white, tinged with Prussian blue, to 
give a blue gray, and carmine also if a purplish gray is desired. 

Remarks. — a. A scale may be used, or not, in making this 
plate. The number of stones shown in the width of the plans, 
shows that the walls are quite thick. 

~b. Bubble walls, not of slate, are, strictly, of two kinds: first, 
those formed of small boulders, used whole, or nearly so ; and 
second, those built of broken rock. Each should show the broad- 
est surfaces in plan. 

c. After tinting, add pen-strokes, called hatchings, to repre- 
sent the character of the surface; as in Fig. A., for rough, or un- 
dressed stone; in waving rows from left to right of short, fine, 
equal, vertical strokes, for smooth stone; and in a mixture of 
numerous fine dots and small angular marks, for a finely picked- 
up surface. (See actual stone work, good drafting copies, and 
mv " Drafting Instruments and Operations.") 



PJL.VT 






(c) 



IE) 




CHAPTER H. 

OONSTEITCTIONS IN WOOD 

§ I. — General Remarks. 

80. Two or more beams may be framed together, so as to make 
any angle with each other, from 0° to 180° ; and so that the plane 
of two united pieces may be vertical, horizontal, or oblique. 

81. To make the present graphical study of framings more fully 
rational, it may here be added, that pieces may be framed with 
reference to resisting forces which would act to separate them in 
the direction of any one of the three dimensions of each. Follow- 
ing out the classification in the preceding article, let us presently 
proceed to notice several examples, some mainly by general de- 
scription of their material construction and action, and some by a 
complete description of their graphical construction and execution^ 
also. 

82. Two other points, however, may here be mentioned. First: 
A pair of pieces may be immediately framed into each other, or 
they may be intermediately framed by "bolts," "keys," &c, or 
both modes may be, and often are, combined. Second: Two com- 
binations of timbers which are alike in general appearance, may be 
adapted, the one to resist extension, and the other, compression, 
and may have slight corresponding differences of construction. 

83. Note. — For the benefit of those who may not have had access 
to the subject, the following brief explanation of scales, &c, is here 
inserted. (See my "Drafting Instruments and Operations.") 

Drawings, showing the pieces as taken apart so as to show the 
mode of union of the pieces represented, are called "Details" 

Sections, are the surfaces exposed by cutting a body by planes, 
and, strictly, are in the planes of section. 

Sectional elevations, or plans, show the parts both in, and be- 
yond, the planes of section. 

Drawings are made in plan, side and end elevations, sections and 
details, or in as few of these as will show clearly all parts of the 
object represented. 

84. In respect to the instrumental operations, these drawings are 



40 CONSTRUCTIONS £N WOOD. 

supposed to be " made to scale," from measurements of models, ot 
from assumed measurements. It will, therefore, be necessary, 
before beginning the drawings, to explain the manner of sketching 
the object, and of taking and recording its measurements. 

85. In sketching the object, make the sketches in the same way 
in which they are to be drawn, i.e. in plan and elevation, and not 
in perspective, and make enough of them to contain all the mea- 
surements, i.e. to show all parts of the object. 

In measuring, take measurements of all the parts which are to be 
shown ; and not merely of individual parts alone, but such con- 
necting measurements as will locate one part with reference to 
another. 

86. The usual mode of recording the measurements, is, to indi- 
cate, by arrow heads, the extremities of the line of which the figures 
between the arrow heads show the length. 

87. For brevity, an accent (') denotes feet, and two accents (") 
denote inches. The dimensions of small rectangular pieces are 
indicated as in PL VII. , Fig. 50, and those of small circular pieces, 
as in Fig. 51. 

88. In the case of a model of an ordinary house framing, such as 
it is useful to have in the drawing room, and in which the sill is 
represented by a piece whose section is about 2^- inches by 3 inches, 
a scale of one inch to six inches is convenient. Let us then describe 
this scale, which may also be called a scale of two inches to the 
foot. 

The same scale may also be expressed as a scale of one foot to 
two inches, meaning that one foot on the object is represented by 
two inches on the drawing ; also, as a scale of £, thus, a foot being 
equal to twelve inches, 12 inches on the object is represented by 
two inches on the drawing ; therefore, one inch on the drawing 
represents six inches on the object, or, each line of the drawing is 
\ of the same line, as seen upon the object ; each line, for we know 
from Geometry that surfaces are to each other as the squares of 
their homologous dimensions, so that if the length of the lines of 
the drawing is one-sixth of the length of the same lines on the object, 
the area of the drawing would be one thirty-sixth of the area of the 
object, but the scale always refers to the relative lengths of the 
lines only. 

89. In constructing the scale above mentioned, upon the stretched 
drawing paper, see PI. VI. , Fig. B. 

1st. Set on upon a fine straight pencil line, two inches, say three 
times, making four points of division. 



CONSTRUCTIONS IN WOOI>. 41 

2d. Number the left hand one of these points, 12, the next, 0, 
the next, 1, the next, 2, &c, for additional points. 

Zd. Since each of these spaces represents a foot, if any one of 
them, as the left hand one, be divided into twelve equal parts, 
those parts will be representative inches. Let the left hand space, 
from (12) to (0) be thus divided, by fine vertical dashes, into twelve 
equal parts, making the three, six, and nine inch marks longer, so 
as to catch the eye, when using the scale. 

4th. As some of the dimensions of the object to be drawn are 
measured to quarter inches, divide the first and sixth of the inches, 
already found, into quarters ; dividing two of them, so that each 
may be a check upon the other, and so that there need be no con 
tinual use of one of them, so as to wear out the scale. 

5th. When complete, the scale may be inked ; the length of it in 
fine parallel lines about ^ of an inch apart. 

90. It is now to be remarked that these spaces are always to be 
called by the names of the dimensions they represent, and not 
according to their actual sizes, i. e. the space from 1 to 2 repre- 
sents a foot upon the object, and is called a foot; so each twelfth 
of the foot from 12 to is called an inch, since it represents an 
inch on the object; and so of the quarter inches. 

91. Next, is to be noticed the directions in which the feet and 
inches are to be estimated. 

The feet are estimated from the zero point towards the right, 
and the inches from the same point towards the left. 

Thus, to take off 2' — 5" from the scale, place one leg of the 
dividers at 2, and extend the other to the fifth inch mark beyond 
0, to the left ; or, if the scale were constructed on the edge of a 
piece of card-board, the scale being laid upon the paper, and, with 
its graduated edge against the indefinite straight line on which the 
given measurement is to be laid off, place the 2' or the 5" mark, at 
that point on the line, from which the measurement is to be laid 
off, according as the given distance is to be to the left or right 
of the given point, and then with a needle point mark the 5" point 
or the 2' point, respectively, which will, with the given point, 
include the required distance. 

92. Other scales, constructed and divided as above described, 
only smaller, are found on the ivory scale, marked 30, <fec, mean- 
ing 30 feet to the inch when the tenths at the left are taken as 
feet; and meaning three feet to the inch when the larger spaces 
— three of which make an inch — are called feet, and the twelfths 
of the left hand space, inches. Intermediate scales are marked 



42 CONSTRUCTIONS IN WOOD. 

35, etc. Thus, on the scale marked 45, four and a half of the 
larger spaces make one inch, and the scale is therefore one of four 
and a half feet to one inch, when these spaces represent feet ; 
and of forty-five feet to one inch, when the tenths represent feet. 
In like manner the other scales may be explained. 

So, on the other side of the ivory, are found scales marked -§. 
&c, meaning scales of ■§■ inch to one foot, or ten feet, according as 
the whole left hand space, or its tenth, is assumed as representing 
one foot. Note that ■§ of an inch to a foot is -| of a foot to the 
inch, ■§- of an inch to ten feet, is 16 feet to an inch, &c. 

94. Of the immense superiority of drawing by these scales, over 
drawing without them, it is needless to say much : without them, 
we should have to go through a mental calculation to find -the 
length of every line of the drawing. Thus, for the piece which is 
two and a half inches high, and drawn to a scale of two inches to 

a foot, we should say — 2-J inches =ff of a foot = 2 5 * °f a f° ot * 
One foot on the object=two inches on the drawing, then ^ of a 
foot on the object =^j of 2 inches =^-= T i g . of an inch, and / T of a 
foot (=2£ inches) =-g T of 2 inches = T 5 ¥ of an inch. 

A similar tedious calculation would have to be gone through 
with for every dimension of the object, while, by the use of scales, 
like that already described, we take off the same number of the feet 
and inches of the scale, that there are of real feet and inches in any 
given line of the object. 

§ II. — Pairs of Timbers whose axes make angles o/0° with each 

other. 

The student should be required to vary all of the remaining con- 
structions in this Division, in one or more of the following ways. 
First, by a change of scale; Second, by choosing other examples 
from models or otherwise, but of similar character; or, Third, by 
a change in the number and arrangement of the projections em- 
ployed in representing the following examples. 

95. Example 1. A Compound Beam bolted. PL VII. , Fig. 46. 
Mechanical Construction. 

The figure represents one beam as laid on top of another. Thus 
situated, the upper one may be slid upon the lower one in the 
direction of two of its three dimensions ; or it may rotate about 
any one of its three dimensions as an axis. A single bolt, passing 
through both beams, as shown in the figure, will prevent all of 
these movements except rotation about the bolt as an axis. Two 



CONSTRUCTIONS IN WOOD. 43 

or more bolts will prevent this latter, and consequently, all move- 
ment of either of the beams upon the other. A bolt, it may be 
necessary to say, is a rod of iron whose length is a little greater 
than the aggregate thickness of the pieces which it fastens toge- 
ther. It is provided at one end with a solid head, and at the other, 
with a few screw threads on which turns a "nut," for the purpose 
of gradually compressing together the pieces through which the 
bolt passes. 

96. Graphical Construction. Assuming for simplicity's sake in 
this and in most of these examples, that the timbers are a foot square^ 
and having the given scale; the diagrams will generally explain 
themselves sufficiently. The scales are expressed fractionally, adja- 
cent to the numbers of the diagrams. The nut only is shown in the 
plan of this figure. 

It is an error to suppose that the nuts and other small parts can 
be carelessly drawn, as by hand, without injury to the drawing, 
since these parts easily catch the eye, and if distorted, or roughly 
drawn, appear very badly. 

The method is, therefore, here fully given for drawing a nut 
accurately. Take any point in the centre line, ab, of the bolt, pro- 
duced, and through it draw any two lines, cd and en, at right-angles 
to each other. From the centre, lay off each way on each line, half 
the length of each side of the nut, say £ of an inch. 

Then, through the points so found, draw lines parallel to the 
centre lines cd and en, and they will form a square plan of a nut 
l£" on each side. 

In making this construction, the distances should be set off very 
carefully, and the sides of the nut ruled, in very fine lines, and 
exactly through the points located. From the plan, the elevation 
is found as in PI. II., Fig. 21. 

97. Ex. 2. A Compound Beam, notched and bolted. PI. 
VII., Fig. 47. Mechanical Construction. The beams represented in 
this figure, are indented together by being alternately notched ; the 
portions cut out of either beam being a foot apart, a foot in length, 
and two inches deep. When merely laid., one upon another, they 
will offer resistance only to being separated longitudinally, and to 
horizontal rotation. 

The addition of a bolt renders the " compound beam," thus 
formed, capable of resisting forces tending to separate it in all 
ways. 

Thin pieces are represented, in this figure, between the bolt-head 
and nut, and the wood. These are circular, having a rounded 



44 CONSTRUCTIONS IN WOOD. 

edge, and a circular aperture in the middle through which the bolt 
passes. They are called " washers" and their use is, to distribute 
the pressure of the nut or bolt-head over a larger surface, so as not 
to indent the wood, and so as to prevent a gouging of the wood 
in tightening the nut, which gouging would facilitate the decay 
of the wood, and consequently, the loosening of the nut. 

98. Graphical Construction. — The beams being understood to be 
originally one foot square, the compound beam will be 22 inches 
deep ; hence draw the upper and lower edges 22 inches apart, and 
from each of them, set off, on a vertical line, 10 inches. Through 
the points, a and 5, so found, draw very faint horizontal lines, and 
on either of them, lay off any number of spaces ; each, one foot in 
length. Through the points, as c, thus located, draw transverse 
lines between the faint lines, and then, to prevent mistakes in 
inking, make slightly heavier the notched line which forms the real 
joint between the timbers. 

The use of the scale of 2V continues till a new one is mentioned. 

The following empirical rules will answer for determining the 
sizes of nuts and washers on assumed sketches like those of PI. VII., 
so as to secure a good appearance to the diagram. The side of the 
nut may be double the diameter of the bolt, and the greater dia- 
meter of the washer may be equal to the diagonal of the nut, plus 
twice the thickness of the washer itself. 

Execution. — This is manifest in this case, and in most of the fol- 
lowing examples, from an inspection of the figures. 

99. Ex. 3. A Compound Beam, keyed. PI. VII.. Fig. 48. 
Mechanical Construction. The defect in the last construction is, 
that the bearing surfaces opposed to separation in the direction 
of the length of the beam, present only the ends of the grain to 
each other. These surfaces are therefore liable to be readily 
abraded or made spongy by the tendency to an interlacing action 
of the fibres. Hence it is better to adopt the construction given 
in PI. VII., Fig. 48, where the "keys," as K, are supposed to be 
of hard wood, whose grain runs in the direction of the width of the 
beam. In this case, the bolts are passed through the keys, to pre- 
vent them from slipping out, though less boring would be required 
if they were placed midway between the keys. In this example, 
the strength of the beam is greatly increased with but a very 
Bmall increase of material, as is prcved in mechanics and confirmed 
by experiment. 

100. Graphical Construction. — This example differs from the last 
jo slightly as to render a particular explanation unnecessary. The 



CONSTRUCTIONS IN WOOD. 45 

keys are 12 inches in height, and 6 inches in width, and are 18 
inches apart from centre to centre. They are most accurately 
located by their vertical centre lines, as AA\ If located thus, and 
from the horizontal centre line BB', they can be completely drawn 
before drawing ee' and nn'. The latter lines, being then pencilled, 
only between the keys, mistakes in inking will be avoided. 

Execution. — The keys present the end of their grain to view, 
hence are inked in diagonal shade lines, which, in order to render 
the illuminated edges of the keys more distinct, might terminate, 
uniformly, at a short distance from the upper and left hand edges. 

By shading only that portion of the right hand edge of each key, 
which is between the timbers, it is shown that the keys do no* 
project beyond the front faces of the timbers. 

101. Ex. 4. A Compound Beam, scarfed. PL VII., Fig. 49. 
Mechanical Construction. This specimen shows the use of a series 
of shallow notches in giving one beam a firm hold, so to speak, 
upon another ; as one deep notch, having a bearing surface equal to 
that of the four shown in the figure, would so far cut away the 
lower beam as to render it nearly useless. 

102. Graphical Construction. — The notches, one foot long, and 
two inches deep, are laid down in a manner similar to that described 
under Ex. 2. 

103. Execution. — The keys, since they present the end of the grain 
to view, are shaded as in the last figure. Heavy lines on their right 
hand and lower edges would indicate that they projected beyond 
the beam. 

Remark. — When the surfaces of two or more timbers lie in the 
same plane, as in many of these examples, they are said to be 
" flush?' with each other. 

§ III. — Combinations of Timbers, whose axes make angles of 90° 
with each other. 

104. The usual way of fastening timbers thus situated, is by 
means of a projecting piece on one of them, called a " tenon," 
which is inserted into a corresponding cavity in the other, called a 
" mortise." The tenon may have three, two, or one of its sides 
flush with the sides of the timber to which it belongs ; while the 
mortise may extend entirely, or only in part, through the timber 
in which it is made, and may be enclosed by that timber on three 
or on all sides. [See the examples which follow, in which some of 
these cases are represented, and from which the rest can be under 
stood.] 



46 CONSTRUCTIONS IX WOOD. 

When the mortise is surrounded on three or on two sides, par- 
tieularly in the latter case, the framed pieces are said to be 
"halved" together, more especially in case they are of equal thick- 
ness, and have half the thickness of each cut away, as at PL VII., 
Fig. 52. 

105. Example 1. Two examples of a Floor Joist and Sill. 
(From a Model.) PI. VII., Fig. 53. Mechanical Construction. 
A — A' is one sill, B — B' another. CC is a floor timber framed into 
both of them. At the left hand end, it is merely " dropped in," with 
a tenon ; at the right hand end, it is framed in, with a tenon and 
" tusk," e. At the right end, therefore, it cannot be lifted out, 
but must be drawn out of the mortise. The tusk, e, gives as great 
a thickness to be broken off, at the insertion into the sill, and as 
much horizontal bearing surface, as if it extended to the full depth 
of the tenon, t, above it, while Jess of the sill is cut away. Thus, 
labor and the strength of the sill, are saved. 

106. Graphical Construction. — 1st. Draw ab. 2d. On ab con- 
struct the elevation of the sills, each 2\ inches by 3 inches. 3d. 
Make the two fragments of floor timber with their upper surfaces 
flush with the tops of the sills, and 2 inches deep. 4th. The mor- 
tise in A', is f of an inch in length, by 1 inch in vertical depth. 
5th. Divide cd into four equal parts, of which the tenon and tusk 
occupy the second and third. The tenon, t, is f of an inch long, 
and the tusk, e, £ of an inch long. Let the scale of £ be used. 

107. Execution. — The sills, appearing as sections in elevation, are 
shaded. In all figures like this, dotted lines of construction should 
be freely used to assist in " reading the drawing," i.e. in com- 
prehending, from the drawing, the construction of the thing repre- 
sented. 

108. Ex. 2. Example of a "Mortise and Tenon," and of 
"Halving." (From a Model.) PI. VII., Fig. 54. Mechanical 
Construction. In this case, the tenon, AA', extends entirely 
through the piece, CC, into which it is framed. B and C are 
halved together, by a mortise in each, whose depth equals half 
the thickness of B, as shown at B" and 0", and by the dotted 
line, ab. 

Graphical Construction. — Make, 1st, the elevation, A ; ; 2d, the 
plan ; 3d, the details. B" is an elevation of B as seen when 
looking in the direction, BA. C is an elevation of the left hand 
portion of CC, showing the mortise into which B is halved. The 
dimensions may be assumed, or found by a scale, as noticed below 

109. Execution. — The invisible parts of the framing, as the halv 



CONSTRUCTIONS IN WOOD. 4? 

ing, as seen at ah in elevation, are shown in dotted lines. The 
brace and the dotted lines of construction serve to show what 
separate figures are comprehended under the general number (54) 
of the diagram. The scale is ±. From this the dimensions of the 
pieces can be found on a scale. 

110. Ex. 3. A Mortise and Tenon as seen in two sills 
and a post. Use of broken planes of section. (From a 
Model.) PL VII. , Fig. 55. 

Mechanical Construction. — The sills, being liable to be drawn 
apart, are pinned at a. The post, BB', is kept in its mortise, W\ by 
its own weight ; m is the mortise in which a vertical wall joist 
rests. It is shown again in section near m'. 

111. Graphical Construction. — The plan, two elevations, and a 
broken section, show all parts fully. 

The assemblage is supposed to be cut, as shown in the plan by 
the broken line AA'A"A'", and is shown, thus cut, in the shaded 
figure, A'A'A"W. The scale, which is the same as in Fig. 53, 
indicates the measurements. At B", is the side elevation of the 
model as seen in looking in the direction A' A. 

In Fig. 55 a, A's obviously equals A"s, as seen in the plan. 

112. Mcecution. — In the shaded elevation, Fig. 55a, the cross-sec- 
tion, A A'", is lined as usual. The longitudinal sections are shaded 
by longitudinal shade lines. The plan of the broken upper end of 
the post, B, is filled with arrow heads, as a specimen of a way 
sometimes convenient, of showing an end view of a broken end. 

Sometimes, though it renders the execution more tedious, narrow 
blank spaces are left on shaded ends, opposite to the heavy lines, 
so as to indicate more plainly the situation of the illuminated edges 
(100). The shading to the left of A', Fig. 55a, should be placed 
so as to distinguish its surface from that to the right of A'. 

113. Ex. 4. A Mortise and Tenon, as seen in timbers so 
framed that the axis of one shall, when produced, be a 
diagonal diameter of the other. PL VII. , Fig. 56. Mechani- 
cal Construction. — In this case the end of the inserted timber is not 
square, and in the receiving timber there is, besides the mortise, a 
tetraedron cut out of the body of that timber. 

114. Graphical Construction. — D is the plan, D' the side eleva- 
tion, and D" the end elevation of the piece bearing the tenon. F' 
and F are an elevation and plan of the piece containing the mor- 
tise. Observe that the middle line of D, and of D', is an axis of 
symmetry, and that the oblique right hand edges of D and D' ar« 
parallel to the corresponding sides of the incision in F'. 



4g CONSTRUCTIONS IN WOOD. 

§ IV. — Miscellaneous Combinations. 

115. Example 1. Do welling. (From a Model.) PL TIL, Fig.57. 
Mechanical Construction. — Dowelling is a mode of fastening by 
pins, projecting usually from an edge of one piece into correspond- 
ing cavities in another piece, as seen in the fastening of the parts 
of the head of a water tight cask. The mode of fastening, how- 
ever, rather than the relative position of the pieces, gives the name 
to this mode of union. 

The example shown in PL VII,, Fig. 57, represents the braces of 
a roof framing as dowelled together with oak pins. 

116. Graphical Construction. — This figure is, as its dimensions 
indicate, drawn from a model. The scale is one-third of an inch 
to an inch. 

1st. Draw acb, with its edges making any angle with the imagi 
nary ground line — not drawn. 

2d. At the middle of this piece, draw the pin or dowel, pp, J of 
an inch in diameter, and projecting J of an inch on each side of the 
piece, acb. This pin hides another, supposed to be behind it. 

Sd. The pieces, d and d'', are each 2\ inches by 1 inch, and are 
shown as if just drawn off from the dowels, but in their true direc- 
tion, i.e. at right angles to acb. 

4th. The inner end of d is shown at d\ showing the two holes, 
1 J inches apart, into which the dowels fit. 

Execution. — The end view is lined as usual, leaving the dowel 
holes blank. 

117. Ex. 2. A dovetailed Mortise and Tenon. PL TIL, 
Fig. 58. Mechanical Construction. — This figure shows a species of 
joining called dovetailing. Here the mortise increases in width as 
it becomes deeper, so that pieces which are dovetailed together, 
either at right angles or endwise, cannot be pulled directly apart. 
The corners of drawers, for instance, are usually dovetailed ; and 
sometimes even stone structures, as lighthouses, which are exposed 
to furious storms, have their parts dovetailed together. 

118. Graphical Construction. — The sketches of this framing are 
arranged as two elevations. A bears the dovetail, B shows the 
length and breadth of the mortise, and B" its depth. A and B 
belong to the same elevation. 

Execution. — In this case a method is given, of representing a 
hidden cut surface, viz. by dotted shade lines, as seen in the hidden 
faces of the mortise in B". 

119. Leaving now the examples of pieces framed together at 
right, angles let us consider : — 



P£,VII 




x-v^ v 



B 



A 



T 



^. 



,5c -i 



B ' 



..'''* " M 







N 




D 


^ 




4 


ID 


4 













CONSTRUCTIONS IN" WOOD. 49 



§ V. — Pairs of Timbers which are framed together obliquely tc 

each other. 

Example 1. A Chord and Principal. (From a Model.) PI. 
VIIL, Fig. 59. Mechanical Construction. — The oblique piece 
(" principal ") is, as the two elevations together show, of equal 
width with the horizontal piece ("chord," or "tie beam"), and ia 
framed into it so as to prevent sliding sidewise or lengthwise. 

K either can it be lifted out, on account of the bolt which is made 
to pass perpendicularly to the joint, ac, and is "chipped up" at^>jo, 
so as to give a flat bearing, parallel to ac, for the nut and bolt-head. 

120. Graphical Construction. — 1st. Draw pde ; 2d. Lay off de 
= 13 inches; 3d. Make e'ea=S0°; 4th. At any point, e\ draw a 
perpendicular to ee\ and lay off upon it 9 inches — the perpendicular 
w r idth of e'ea; 5th. Make ec = 4 inches and perpendicular to e'e ; 
bisect it and complete the outlines of the tenons, and the shoulder 
uec' '; Qth. To draw the nut accurately, proceed as in PI. VII., Fig. 
46-47, placing the centre of the auxiliary projection of the nut in 
the axis of the bolt produced, &c. (46) (96). b represents the bolt 
hole, the bolt being shown only on one elevation. 

121. Ex. 2. A Brace, as seen in the angle between a 
"post" and "girth." (From a Model.) PI. VIII., Fig. 60. 
Mechanical Construction. — PP' is the post, GG' is the girth, and 
B'B" is the brace, having a truncated tenon at each end, which rests 
in a mortise. When the brace is quite small, it has a shoulder on 
one side only of the tenon, as if B'B" were sawed lengthwise on 
a line, oo'. 

122. Graphical Construction. — To show a tenon of the brace 
clearly, the girth and brace together are represented as being 
drawn out of the post. 1st. Draw the post. 2d. Half an inch 
below the top of the post, draw the girth 2j inches deep. 3d. From 
«, lay off ab and a each 4 inches, and draw the brace 1 inch wide, 
\th. Make cd equal to the adjacent mortise; viz. 1^- inches; make 
de = l inch, and erect the perpendicular at e till it meets be, etc. 
The dotted projecting lines show the construction of B" and of the 
plan. At e" is the vertical end of the tenon e. On each side of 
e", are the vertical surfaces, shown also at cd. Let B' also be pro- 
jected on a plane parallel to be. 

123. Ex. 3. A Brace, with shoulders mortised into the 
post. PI. Vlll., Fig. 61. Tliis is the strongest way of framing a 
brace. For the rest, the figure explains itself. Observe, however, 
that while in Fig. 59, the head of the tenon and shoulder is perpendi- 



50 CONSTRUCTIONS IN WOOL 

cular to the oblique piece, here, where that piece id framed flito a 
vertical post, its head nu is perpendicular to the axis of the post. 
In Fig. 61, moreover, the auxiliary plane on which the brace 
alone is projected, is parallel to the length of the brace, as is 
shown by the situation of B, the auxiliary projection of the brace, 
and by the direction of the projecting lines, as rr', P' is the 
elevation of P, as seen in the direction nri, and with the brace 
removed. 

124. Ex. 4. A "Shoar." PI. Yin., Pig. 62. Mechanical Con- 
struction. — A " shoar " is a large timber used to prop up earth or 
buildings, by being framed obliquely into a horizontal beam 
and a stout vertical post. It is usually of temporary use, during 
the construction of permanent works ; and as respects its action, 
it resists compression in the direction of its length. To give a 
large bearing surface without cutting too far into the vertical tim- 
ber, \t often has two shoulders. The surface at ab is made verti- 
cal, for then the fibres of the post are unbroken except at cb, while 
if the upper shoulder were shaped as at adb, the fibres of the trian- 
gular portion dbc would be short, and less able to resist a longitu- 
dinal force. 

125. The Graphical construction is evident from the figure, 
which is in two elevations, the left hand one showing the post 
only. 

Execution. — The vertical surface at ab may, in the left hand 
elevation, be left blank, or shaded with vertical lines as in PL VII. , 
Fig. 55a. 

§ VL — Combinations of Timbers whose axes make angles of 180° 
with each other. 

126. Timbers thus framed are, in general, said to be spliced. Six 
forms of splicing are shown in the following figures. 

Example 1. A Halved Splicing, pinned. PL VIII, Fig. 63. 
The mechanical construction is evident from the figure. When 
boards are lapped on their edges in this way, as in figure 69, they 
are said to be " rabbetted." 

127. Graphical Construction. — After drawing the lines, 12 inches 
apart, which represent the edges of the timber, drop a perpendicu- 
lar of 6 inches in length from any point as a. From its lower ex- 
tremity, draw a horizontal line 12 inches in length, and from c, drop 
the perpendicular cb, which completes the elevation. In the plan, 
the joint at a will be seen as a full line a'a\ and that at £, being 
hidden, is represented as a dotted line, at b" 



CONSTRUCTIONS IN WOOD. 5t 

Draw a diagonal, a'b n , divide it into four equal parts, and take 
the first and third points of division for the centres of two pins, 
having each a radius of three-fourths of an inch. 

Execution. — The position of the heavy lines on these figures is 
too obvious to need remark. 

128. Ex. 2. Tonguing and Grooving; and Mortise and 
Tenon Splicing. PI. VIII. , Fig. 64. Boards united at their 
edges in this way, as shown in PI. VIII., Fig. 70, are said to be 
tongued and grooved. 

Drawing, as before, the plan and elevation of a oeam, a foot 
square, divide its depth an, at any point a, into five equal parts. 
Take the second and fourth of these parts as the width of the 
tenons, which are each a foot long. 

The joint at a is visible in the plan, the one at b is uut. Let a'd 
be a diagonal line of the square a'd. Divide a'd into three equal 
parts, and take the points of division as centres of inch bolts, with 
heads and nuts 2 inches square, and washers of If inches radius. To 
place the nut in any position on its axis, draw any two lines at right 
angles to each other, through each of the bolt centres, and on each, 
lay off 1 inch from those centres, and describe the nut. Project up 
those angles of the nut which are seen ; viz. the foremost ones, 
make it 1 inch thick in elevation, and its washer \ an inch thick. 

In this, and all similar cases, the head of the bolt, s, would not 
have its longer edges necessarily parallel to those of the nut. To 
give the bolt head any position on its axis, describe it in an auxiliary 
plan just below it. 

129. Ex. 3. A Scarfed Splicing, strapped. PI. VIII. , Fig. 65. 
Mechanical Construction. — While timbers, framed together as in 
the two preceding examples, can be directly slid apart when their 
connecting bolts are removed ; the timbers, framed as in the present 
example, cannot be thus separated longitudinally, on account of the 
dovetailed form of the splice. Strapping makes a firm connexion, 
but consumes a great deal of the uniting material. 

130. Graphical Construction. — After drawing the outlines of the 
side elevation, make the perpendiculars at a and a', each 8 inches 
long, and make them 18 inches apart. One inch from a, make the 
strap, ss\ 2£ inches wide, and projecting half an inch — i. e. its 
thickness — over the edges of the timber. The ear through which 
the bolt passes to bind the strap round the timber, projects two 
inches above the strap. Take the centre of the ear as the centre 
of the bolt, and on this centre describe the bolt head l£ inches 
square. 

4 



02 CONSTRUCTIONS IX WOOD. 

In the plan, the ears of the strap are at any indefinite distance 
apart, depending on the tightness of the nut, s. 

A fragment of a similar strap at the other end of the scarf, is 
shown, with its visible ends on the bottom, near a\ and back, at r 
of the beam. 

Execution. — The scarf is dotted where it disappears behind the 
strap ; and so are the hidden joint at a', and the fragment of the 
Becond strap, as shown in plan. These examples may advan- 
tageously be drawn by the student on a scale of y 1 ^. Care must 
then be exercised in making the large broken ends neatly, in large 
splinters, edged with fine ones. 

131. Ex.4. A Scarfed Splice, bolted. PL VIII.. Fig. 6C. 
Mechanical Construction. — AA' is one timber. BB' is the other. 
Each is cut off as at aea", forming a pointed end which prevents 
lateral displacement, fc/and dg are the ends of transverse keys, 
which afford good bearing surfaces. See the same on PL XIII., 
Pig. 105, which shows the arrangement plainly. 

132. Graphical Constrvction. — Let each timber be 1 foot square 
and let the scale be 1 foot to 1 inch. Draw the outlines accord- 
ingly, and, assuming a', make a'b' = 3 feet; drop the perpendicular 
&V, and draw a line of construction a'c f . From a' lay off 4 inches 
on a'c' and divide the rest oi a'c' into three equal parts, to get the 
size of the equal spaces c'k,fd and gv\ and at c' and h draw per- 
pendiculars to a'c', above it and 2 inches long. Divide a'k into 
two equal parts at d, and from h and d lay off 2 inches on a'c' 
towards a' and complete the keys, as shown, hf above a'c', and dg 
below it; also the joints v'g, dk, etc., of the splice, where a'v' is 
perpendicular to a'c' and 2 inches long. 

Now, in plan, project down a' at a and a" and draw ae and a"e 
at 60° with a"c, and do the same with c', as shown. Project up e 
to e' and draw eV parallel to a'v' till it meets c'g produced. Then 
project down v' at v and t>", and s' at s, and draw sv and sv", 
which, of course, will not be parallel to ae and a"e, since se — s'e' is 
shorter than av — a'v'. Thus av se a"v" — a'v' s'e' is the obliquely 
pointed end of the piece AA'. BB' is similarly pointed, as shown. 

Add the bolts, nuts and washers, nn' and mm', in any convenient 
position, which will complete the construction. 

Execution. — The keys are shaded. The hidden cut surfaces of 
the notches are shaded in dotted shade lines, and the hidden joints 
are dotted. 

133. Ex. 5. A Compound Beam, "with one of the compO' 
Qent beams "fished." PL YIII., Fig. 67. Mechanical Construe* 



CONSTRUCTIONS IN WOOD. Do 

tion. — The mode of union called fishing, consists in uniting two 
pieces, end to end, by laying a notched piece over the joint and 
bolting it through the longer pieces. 

The figure shows this mode as applied to a compound beam, 
i. e. to a beam " built " of several pieces bolted and keyed toge- 
ther. The order of construction is as follows — taking for a scale 
\ of an inch to a foot. 

134. Graphical Construction. — 1st. The outside lines of the plan 
are two feet apart, the outside pieces are each 4^- inches wide, 
and the interior ones 5^ inches ; leaving four inches for the sum 
of the three equal spaces between the four beams. 

2d. Let there be a joint at a'. Lay off 3 feet, each side of a' for 
the length of the "fish." 

3d. The straight side of this piece is let into the whole piece, b, 
two thirds of an inch, and into a\ 1 inch, making its thickness 3 
inches. 

4th. At a\ it is two inches thick, i. e. at a\ the timber, a', is of 
its full thickness. The fish, c, is 2 inches thick for the space of one 
foot at each side of a'. The notches at d and e are each 1 inch 
deep, dd' and ee' each are one foot, and the notches at d' aisd e' 
are each 1 inch deep. The remaining portions of the fish are 1 
foot long, and 3 inches wide. 

5th. Opposite to these extreme portions, are keys, 1 foot by 3 
inches, in the spaces between the other timbers, and setting an equal 
depth into each timber. 

6th. In elevation, only the timber a' is seen — 1 foot deep. Four 
bolts pass through the keys. b'b"—5 feet, and b' is three inches from 
the top, and from kk\ the left hand end of the fish, n'n" =5 feet, 
and n' is 3 inches above the bottom of the timber, and 9 inches from 
kk' . The circular bolt head is one inch in diameter, and its washer 
3£ inches diameter and £ an inch thick. The thickness of the bolt 
head, as seen in plan, is f inch. The nuts, nn, are 1^- inches square, 
and J an inch thick, and the bolts are half an inch in diameter. 

135. The several nuts would naturally be found in various posi- 
tions on their axes. To construct them thus with accuracy, as seen 
in the plan, one auxiliary elevation, as N, is sufficient. N, and its 
centre, may be projected upon as many planes — xy — as there are 
different positions to be represented in the plan, each plane being 
supposed to be situated, in reference to N, as some nut in the plan 
is, to its elevation. Then transfer the points on xy, &c, to the 
outside of the several ww£-washers, placing the projection of the 
centre lines of the bolts in the plan, a<* lines of reference. 



54 ' CONSTRUCTIONS IN "WOOD. 

Execution. — The figure explains itself in this respect. 

136. Ex. 6. A vertical Splice. PI. VIII., Fig. 68. Mecha* 
nical Construction. — This splice is formed of two prongs at oppo- 
site corners of each piece, embraced by corresponding notches in 
the other piece. Thus in the piece B', the visible prong, as seen in 
elevation, is a truncated triangular pyramid whose horizontal base 
is abc — dc\ and whose oblique base is enc — e'n"c" . Besides the 
four prongs, two on each timber, there is a flat surface abfq — caq\ 
well adapted to receive a vertical pressure, since it is equal upon, 
and common to, both timbers. 

137. Graphical Construction. — To aid in understanding this 
combination, an oblique projection is given on a diagonal plane, 
parallel to PQ. 

1st. Make the plan, acfq, with the angles of the interior square 
in the middle of the sides of the outer one. 2c?. Make the distances, 
as ce = 2 inches and draw en, &c. 3c7. Make c'n'^c'n" 15 inches, 
and draw short horizontal lines, n"e\ &c, on which project e, &c., 
after which the rest is readily completed. 

138. Execution. — Observe, in the plans, to change the direction 
of the shade lines at every change in the position of the surface of 
the wood. 



CHAPTER IH 

CONSTRUCTIONS IN METAL. 

139. Example 1. An end view of a Railroad Rail. PL 

VIII. , Fig. 71. Graphical Construction. — 1st. Draw a vertical centre 
line AA', and make AA'=3f inches. 2d. Make A'£=A7>'=2 
inches. 3 d. Make Ac = Ac' = 1 inch. 4th. Describe two quadrants, 
of which c'd is one, with a radius of half an inch. 5th. With p, 
half an inch from AA', as a centre, and pd as a radius, describe an 
arc, dr, till it meets a vertical line through e. 6th. Draw the tangent 
rs. 1th. Draw a vertical line, as pt, % an inch from AA', on each 
side of AA'. 8th. Bisect the angle rst' and note s', where the 
bisecting line meets the radius, pr, produced. 9th. With s' as a 
centre, draw the arc rt'. 10th. At b and b' erect perpendiculars, 
each one fourth of an inch high. 11th. Draw quadrants, as q't, 
tangent to these perpendiculars and of one fourth of an inch radius. 
12th. Draw the horizontal line tv. 13th. Make nt'=nv and describe 
the arc t'v. 14th. Repeat these operations on the other side of the 
centre line, AA'. 

Execution. — Let the construction be fully shown on one side of 
the centre line. 

140. Ex. 2. An end elevation of a Compound Rail. PL VIII., 
Fig. 12. Mechanical Construction.- — The compound rail, is a rail 
formed in two parts, which are placed side by side so as to break 
joints, and then riveted together. As one half of the rail is whole, 
at the points where a joint occurs on the other half, the noise and 
jar, observable in riding on tracks built in the ordinary manner, are 
both obviated ; also " chairs," the metal supports which receive the 
ends of the ordinary rails, may be dispensed with, in case of the 
use of the compound rail. 

In laying a compound rail on a curve, the holes, through which 
the bolts pass, may be drawn past one another by the bending 
of the rails. To allow for this, these holes are " slotted,'' as it is 
termed, i. e. made longer in the direction of the length of the rail. 

141. Graphical Construction. — 1st. Make ty=4 inches. 2d. 
Bisect ty at w, and erect a perpendicular, w«, of 3\ inches. 3d. 



5G CONSTRUCTIONS IN METAL. 

Make, successively, ur=% an inch; fi'om r to cb = 2 inches; from u 
to nh=^ of an inch ; and to ge — 2^ inches. 4th. For the several 
widths of the interior parts, make 5c=§ of an inch, and g and 
each f of an inch, from ua ; nh—ge^ and ro—% of an inch. 5th. 
To locate the outlines of the rail, make ws, the flat top, called the 
tread of the rail, = 2 inches, half an inch below this, make the width, 
fd, 3 inches ; and make the part through which the rivet passes, If 
inches thick, and rounded into the lower flange which is f of an 
inch thick. 

The rivet has its axis If inches from ty. Its original head, q, is 
conical, with bases of — say 1 inch, and f of an inch, diameter ; and 
is half an inch thick. The other head, p, is made at pleasure, being 
roughly hammered down while the rivet is hot, during the process 
of track-laying. A thin washer is shown under this head. 

142. Ex. 3. A " Cage Valve," from a Locomotive Pump. 
PI. VIII., Fig. 73-74. — Mechanical Construction. — This valve is 
made in three pieces, viz. the valve proper, Fig. 74 ; the cage con- 
taining it, Abb' ; and the flange bb'c ; whose cylindrical aperture — - 
shown in dotted lines — being smaller than the valve, confines it. 
The valve is a cup, solid at the bottom, and makes a water tight 
joint with the upper surface of the flange, inside of the cage. The 
whole is inclosed in a chamber communicating with the pump 
barrel, and wifch the tender, or the boiler, according as we suppose 
it to be the inlet or outlet valve of the pump. This chamber makes 
a water tight joint with the circumference of the flange bb.' 

Suppose the valve to be the latter of the two just named. The 
il plunger " of the pump being forced in, the water shuts the inlet 
valve, and raises the outlet valve, and escapes between the bars of 
the cage into the chamber, and from that, by a pipe, into the boiler. 

143. Graphical Construction. — Scale full size. Make the plan 
first, where the six bars are equal and equidistant, with radial sides. 
Project them into the elevation; as in Prob. 14, Div. I.; taking 
care to note whether any of the bars, as E, on the back part of the 
cage can be seen above the valve, CD, and between the front bars, 
as F and G. The diameters of the circles seen in the cage are, in 
order, from the centre, 1£, 2\, 2^|-, and 4 inches. The thickness of 
the valve, Fig. 74, is -^ of an inch, its outside height If inches, 
and the outside diameter 2^\ inches. The diameter of the aperture 
in the flange is If inches, its length J of an inch, and the height of 
the whole cage is 3^- inches. 

144. Execution. — Observe carefully the position of the heavy 
lines. The section, Fig. 74, being of metal, is finely shaded. 



CONSTRUCTIONS IN METAL. 5? 

145. Ex. 4. An oblique elevation of a Bolt Head. PL 
VIII., Fig. 75, Let PQ be the intersection of two vertical planes, 
at right angles with each other; and let RS be the intersection, 
with the vertical plane of the paper to the left of PQ, of a plane 
which, in space, is parallel to the square top of the bolt head. On 
such a plane, a plan view of the bolt head may be made, showing 
two of its dimensions in their real size; and on the plane above RS, 
the thickness of the bolt head, and diameter of the bolt, are shown 
in their real size. 

Below RS, construct the plan of the bolt head, with its sides 
making any angle with the ground line RS. Project its corners in 
perpendiculars to RS, giving the left hand elevation, whose thick- 
ness is assumed. 

146. The fact that the projecting lines of a point, form, in the 
drawings, a perpendicular to the ground line, is but a special case 
of a more general truth, which may be thus stated. — When an 
object in space is projected upon any two planes which are at right 
angles to each other, the projecting lines of any point of that object 
form a line, in the drawing, perpendicular to the intersection of the 
two planes. 

147. To apply the foregoing principle to the present problem ; 
it appears that each point, as a", of the right hand elevation, will 
be in a line, a' a", perpendicular to PQ, the intersection of the two 
vertical planes of projection. 

Remembering that PQ is the intersection of a vertical plane — 
perpendicular to the plane of the paper — with the vertical plane 
of the paper, and observing that the figure represents this plane as 
being revolved around PQ towards the left, and into the plane of 
the paper, and observing the arrow, which indicates the direction 
in which the bolt head is viewed, it appears that the revolved ver- 
tical plane, has been transferred from a position at the left of the 
plan acne, to the position, PQ, and that the centre line tu", must 
appear as far from PQ as it is in front of the plane of the paper— 
i. e. e"V r =eu, showing also, that as e — e' is in the plane of the paper, 
its projection at e" must be in PQ, the intersection of the two ver- 
tical planes. 

Similarly, the other corners of the nut, as c\ n\ <fcc, are laid off 
either from the centre line tu'\ or from PQ. Thus v' f c''=vc, or 
v'"c" = sc. The diameter of the bolt is equal in both elevations. 

148. Other supposed positions of the auxiliary plane PQ may be 
assumed by the student, and the corresponding construction worked 
out. Thus, the primitive position of PQ may be at the right of the 



58 CONSTRUCTIONS IN METAL. 

bolt head, and that may be viewed in the opposite direction from 
that indicated by the arrow. 

149. Ex. 5. A " Step" for the support of an oblique tim 
ber. PI. VIII. , Fig. 76. Mechanical Construction. — It will be 
frequently observed, in the framings of bridges, that there are 
certain timbers whose edges have an oblique direction in a vertical 
plane, while at their ends they abut against horizontal timbers, not 
directly, for that would cause them to be cut off obliquely, but 
through the medium of a prismatic block of wood or iron, so 
shaped that one of its faces, as ab — rib', Fig. 76, rests on the hori- 
zontal timber, while another, as ac — e"d''c", is perpendicular to the 
oblique timber. 

To secure lightness with strength, the step is hollow underneath, 
and strengthened by ribs, rr. The holes, h'h", allow the passage 
of iron rods, used in binding together the parts of the bridge. These 
holes are here prolonged, as at h, forming tubes, which extend partly 
or wholly through the horizontal timber on which the step rests, in 
order to hold the step steadily in its place. 

150. Remark. When the oblique timber, as T, Fig. 76 (a), sets 
into, rather than upon, its iron support S, so that the dotted lines, 
ab and cb, represent the ends of the timber, the support, S, is called 
a shoe. 

151. Graphical Construction. — In the plate, abc is the elevation, 
and according to the usual arrangement would be placed above the 
plan, e"ri'c", of the top of. the step, a'b'e is the plan of the under 
Bide of the step, showing the ribs, &c. A line through nm is a 
centre line for this plan and for the elevation. A line through the 
middle point, r, of mn, is another centre line for the plan of the 
bottom of the step. Having chosen a scale, the position of the 
centre lines, and the arrangement of the figures, the details of the 
construction may be left to the student. 

152. Ex. 6. A metallic steam tight "Packing," for the 
"stuffing boxes" of piston rods. PI. VIII., Fig. 77. 

Mechanical Construction. — Attached to that end, b, of Fig. 77 (a), 
of a steam cylinder, for instance, at which the piston rod, p, entera 
it, is a cylindrical projection or " neck," n, having at its outer end 
a flange, ff, through which two or more bolts pass. At its inner 
end, at p, this neck fits the piston rod quite close for a short space. 
The internal diameter of the remaining portion of the neck is suffi- 
cient to receive a ring, rr, which fits the piston rod, and has on its 
outer edge a flange, t, by which it is fastened to the flange, ff, on 
the neck of the cylinder by screw bolts. The remaining hollow 



CONSTRUCTIONS IN METAL. 09 

space, s, between the ring or " gland," tr, and the inner end of the 
neck, is usually filled with some elastic substance, as picked hemp, 
which, as held in place by the gland, tr, makes a steam tight joint ; 
which, altogether, is called the " stufling box." 

153. The objection to this kind of packing is, that it requires so 
frequent renewals, that much time is consumed, for instance in rail- 
road repair shops, in the preparation and adjustment of the packing 
To obviate this loss of time, and perhaps because it seems more 
neat and trim to have all parts of an engine metallic, this metallic 
packing, Fig. 77, was invented. ABC — A'B' is a cast iron ring, 
cylindrical on the outside, and having inlaid, in its circumference, 
bands, tt\ of soft metal, so that it may be squeezed perfectly tight 
into the neck of the cylinder. The inner surface of this ring is coni- 
cal, and contains the packing of block tin. This packing, as a whole, 
is also a ring, whose exterior is conical, and fits the inner side of the 
iron ring, and whose interior, fkc, is cylindrical, fitting the piston 
rod closely. 

154. For adjustment, this tin packing is cut horizontally into 
three rings, and each partial ring is then cut vertically, as shown in 
the figure, into two equal segments, abed — a'b'c'd', is one segment 
of the upper ring; efgh — e'f'f"g'h\\s one segment of the middle ring, 
and rjJcl — r'j'k'n'l, is one segment of the lower ring. The segments 
of each ring, it therefore appears, break joints with each of the other 
rings. Three of the segments, one in each partial ring, are loose, 
while the other three are dowelled by small iron pins, parallel to 
the axis of the whole packing. 

155. Operation. — Suppose the interior, fkc, of the packing to be 
of less diameter than the piston rod, which it is to surround. By 
drawing it partly out of its conical iron case, the segments forming 
each ring can be slightly separated, making spaces at ab, &c, which 
will increase the internal diameter, so as to receive the piston rod. 
When in this position, let the gland, tr, be brought to bear on the 
packing, and it will be firmly held in place; then, as the packing 
gradually wears away, the gland, by being pressed further into the 
neck, will press the packing further into its conical seat, which 
will close up the segments round the piston rod. 

156. Graphical Construction. — Let the scale be from one half 
to the whole original size of the packing for a locomotive valve 
chest. C is the centre for the various circles of the plan, and DD', 
projected up from C, is a centre line for portions of the elevation 
0/=^ of an inch; Ce=l^- inches; CA=1J- inches. A'n=2 
inches, and A's'—^ of an inch. The iron case being constructed 



60 



CONSTRUCTIONS IN METAL. 



from, these measurements, the rings must be located so that f'p', tot 
instance, shall be = ,"L. of an inch ; and then let the thickness, ff\ 
of each segment be \\ of an inch. These dimensions and the con- 
sequent arrangements of the rings will give spaces between the seg- 
ments, as at ab, of £ of an inch ; though in fact, as this space is 
variable, there is no necessity for a precise measurement for it. In 
the plan, there are shown one segment, and a fragment, abef, of 
another, in the upper ring ; one segment and two fragments of the 
middle ring, and both segments of the lower ring, with the whole 
of the iron case. The elevation shows one of the halves of each 
ring, viz., acd — a'c'd', the upper one; ef gh — d'f cfh\ the middle 
one ; and rjkl — r'j'k'V, the lower one. 

The circles of the plan are found by projecting the points as p f 
upon the diameter AC. Then by comparing kk' , IV, and nn\ for 
example, we see how the vertical projections, as k'l'n', of the ends 
of the half rings are found. 

157. Execution. — The section lines in the elevation indicate 
clearly the situation of the three segments, abed, efgh, and ijkU 
there shown. The dark bands on the case at t and t' ', indicate the 
inlaid bands of soft metal already described. 

The student can usefully multiply these examples by constructing 
plans, elevations and sections, from measurement, of such objects 
as steam, water, and gas cocks, valves, or gates, railway joints, 
and chairs, and other like simple metallic details; actual examples 
of which can be easily obtained as models almost anywhere. 

Ex. 7. Rolled Iron Beams and Columns. 

Wood is perishable and growing scarce. Stone is comparatively 
costly and cumbrous. Cast iron is suspected as treacherous. 





Fig. 1, Fig. 2. 

Hence, of late years, much attention has been paid to beams and 
columns of rolled wrought iron. Various figures of such work 
are therefore given as an appropriate concluding general example 
for the present chapter. 



PL.V11I. 













! 



rii vlU I 



OK 



.11 


1 


r 

1 


i 


p- 



® 




CONSTRUCTIONS IN METAL. 



61 



Rolled iron in its elementary commercial forms, for architec- 
tural and engineering purposes, is principally known as beam, 
angle, T (Fig. 1), V (Fig. 2), and channel iron, the latter 




Fig. 3. 





Fig. 4. 



Fig. 5. 



either in polygonal or circular segments (Fig. 3). All of these 
are used in building up compound beams, braces, or columns, 
Figs. 4_10, of those here given,* may all be taken as on a scale 





Fig. 6, 



Fig. 7. 



of one fourth the full size, and Fig. 9, as a practical example 
of oblique projection (see Div. IV.), of one eighth the full size. 



* From the Union Iron Works, Buffalo, K Y. 



62 



CONSTRUCTIONS IN METAL. 



Figs. 4-13, except 9, are all transverse sections of the beams or 
columns which, they represent. 




Fig. 8. 

Simple beams being made of all sizes from four -to fifteen inches 
in depth, and some of the sizes either light or heavy, Figs. 4-8 
represent compound beams of depths greater than fifteen inches. 




Fig. 4 represents the lower half of a beam formed of riyeted 
plate and angle irons. 



CONSTRUCTIONS IN METAL. 



63 




Fig. 10. 




Fig. 11. 




Fig. 12. 



64 CONSTRUCTIONS IN METAL. 

Fig. 5 represents the left-hand half of a hollow beam of hori- 
zontal plate, and vertical rectangular channel iron. 

Fig. 6 shows the lower end of a very deep beam, the ends of 
which are symmetrically placed, and formed of plate and curved 
channel iron. 

Fig. 7 is the lower half of a beam composed of a simple beam 
riveted to horizontal plates. 

Fig. 8 represents the lower half of a beam, 16" wide at base, 
composed of plate, beam, and rectangular channel irons. 

Fig. 9, an oblique projection, given here for convenience in 
anticipation of Div. IV. , shows how beams at right angles to each 
other may be riveted together by angle plates. 

Leaving beams for the highly interesting and practically very 
important subject of iron columns, Fig. 10 is the partial hori- 
zontal section of a column curiously formed of two flanged beams, 
bent at right angles, and riveted along the angle, through an 
intermediate doubly concave bar, which gives a firmer bearing 
for the rivets. 

Fig. 11 is nearly a half section of a column composed of six 
curved channel irons, disposed with the convexities inwards, and 
riveted in the angles of the flanges. 

Fig. 12 shows two examples of the true hollow column * as 
made in segments, consisting of channel irons whose flanges are 
radial and which are placed with their convexities outward, and 
are riveted through the flanges. The inner figure represents a 
column of four channel irons, the outer one a larger column of six 
flanges ; the number varying from four to eight in different cases. 

Fig. 13 is a section of a column of German design, made of 
plate and polygonal channel irons. 

The ideal of Fig. 12 is a smooth hollow column in one piece, 
a given amount of matter having much greater strength, in this 
form, to resist crushing or bending than when in a solid bar. 
But this ideal, though easily realized in cast iron, is economically 
impracticable in wrought iron; hence the external radial flanges, 
though giving additional strength by increasing the average dis- 
tance of the material from the centre, are subordinate to the 
main idea. 

In Fig. 11, on the contrary, the heavy flanges are the primary 
feature as a means of gaining strength by a circumferential dis- 
* Made at Phoenixville, Pa. 



CONSTRUCTIONS IN METAL. 



65 



position of material, while the curved parts form a meang of 
connecting them, and of also leaving a hollow interior. 

A plate alone easily "buckles" m the direction of its thick- 
ness. Each half of Fig. 10 is therefore a plate, braced to prevent 
this by bending at right angles, and by means of flanges which 
also make the circumferential strength prominent. 

Fig. 13 is also essentially a braced plate column, the primary 




Fig. 13. 

office of the channel irons bein^ to unite the plates, leaving their 
effect on the circumferential strength incidental. 

If a column could be formed by placing the channel irons of 
Fig. 11 so as to join them by their flanges, which, in Fig. 12, 
would then radiate inward from the circumference of the column, 
this circumference might then be of slightly greater diameter 
for a given amount of matter. But the riveting would be 
inaccessible. It may be a question, however, 
whether collars might not be clamped or 
shrunk on, numerous enough to make the 
column essentially solid, and at no more than 
the cost of riveting. This question appears 
to have exercised other minds, and Fig. 14 
indicates one solution of it,* consisting of 
cylindrical segments bearing dovetailed in 
place of rectangular flanges on their edges, 
and. united by clamping bars in place of riv- 
ets ; as one would hold two boards laid one 
upon the other, by clasping each hand over the 
two edges of the boards, while the riveting plan may be repre- 
sented by thrusting the fingers through holes near the edges of 
the boards. 




* Manufactured by Carnegie Brothers & Co., Pittsburg, Pa. 



DIVISION THIED. 

ELEMENTARY SHADOWS AND SHADING. 



CHAPTER I. 

SHADOWS. 

§ I. — Facts, Principles, and Preliminary Problems. 

158. The shadow of a given opaque body, B, PL IX., Fig. 
78, upon any suriace S, is the portion of S from which light is 
excluded by B. 

A shadow is known when its boundary, sph, called the line of 
shadow, is known. Hence, to find the shadow of a given body 
upon a given surface is, practically, to find the boundary of that 
shadow. 

159. The boundary, sph, PL IX., Fig. 78, of the shadow of a 
body, B, is the shadow of the line of shade, hrn, which divides the 
illuminated from the unilluminated part of B. For if a ray, od, 
pierces S, as at d, within the area of the shadow, it must pierce the 
body B in its illuminated part as at o; but if another ray pierces 
S, as at q, without the shadow, it cannot meet the body B. Hence 
rays, as hs, cp, nh, which meet S in the line of shadow, must be 
tangent to B at points, as h, c, n, of its line of shade. 

160. Since the line of shadow of any body B on any surface S 
is the shadow of the line of shade of B, the line of shade on the 
body casting a required shadow must, in general, be found first 
in problems of shadows. 

On plane-sided, that is flat-sided bodies, this line of shade con- 
sists simply of the edges which divide its faces in the light from 
those in the dark; that is, it consists of the shade lines (21) of the 
body. These may in many cases be found by simple inspection. 

161. Rays of light are here assumed to he parallel straight lines, 
as they practically are when proceeding from a very distant 
source, as the sun, to any terrestrial object. 

1st. It will be thus observed that the shadow of a vertical edge 
ab, PL IX., Fig. 78, of the body of the house will be a vertical 
line, a'b', on the front wall of the wing behind it; that the shadow 



SHADOWS. 67 

of a horizontal line, as be — the arm for a swinging sign — which is 
parallel to the wing wall, will be a horizontal line, b'e', parallel 
to be; that a horizontal line, be, which is perpendicular to the wing 
wall, will have an oblique shadow, cb' , on that wall, commencing 
at c, where the line pierces the wing wall, and ending at b', where 
a ray of light through b pierces the wing wall; and finally that 
the shadow of a point, b, is at b' where the ray bb' > through that 
point, pierces the surface receiving the shadow. 

2c?. Passing now to PL IX. , Fig. 79, which represents a chim- 
ney upon a flat roof, we observe that the shadows of be and cd— 
lines parallel to the roof — are b'c' and c'd', lines equal to, and 
parallel to, the lines be and cd; and that the shadows of ab and ed 
are ab' and ed' — similar to the shadow cb' m Fig. 78 — i. e. com- 
mencing at a and e, where the lines casting them meet the roof, 
and ending at b' and d', where rays through b and d meet the roof. 

162. The facts just noted may be stated as elementary general 
principles by means of which many simple problems can be solved. 

1st. The shadow of a point on any surface, is where a ray of 
light through that point meets that surface. 

2d. The shadow of a straight line, upon a plane parallel to it, 
is a parallel straight line. 

3d. In like manner, the shadow of a circle upon a plane parallel 
to it is an equal circle ; whose centre, only, therefore need be found. 

Uh. The shadow of a line upon a plane to which it is perpen- 
dicular, will coincide with the projection of a ray of light upon that 
plane. Thus ba, Fig. 78, being perpendicular to H, its shadow 
upon that plane coincides with the horizontal projection, aa', of 
the ray of light bb' . Likewise be, being perpendicular to the 
vertical plane cb'a' , its shadow coincides with cb' , the projection 
of bb' on that plane. 

5th. The shadow of a line upon a surface may be said to begin 
where that line meets that surface, either or both being produced, 
if necessary. See a, Fig. 79. 

6th. When the shadow, as aa'b' , of a line, &3 ab, Fig. 78, falls 
upon two surfaces which intersect, the partial shadows, as aa' 
and b'a' , meet this intersection at the same point, as a'. 

1th. The shadow of a straight line, upon a curved surface, or 
of a circle, otherwise than as in (3d) will generally be a curve, 
whose points, separately found by (1st), must then be joined. 



68 SHADOWS. 

163. In applying these principles to the construction of shad- 
ows, three things must evidently be given, viz. — 

1st. The body casting the shadow. 

2d. The surface receiving the shadow. 

3d. The direction of the light. 

And these must be given, not in reality, but in projection. 

164. The light is, for convenience and uniformity, usually as- 
sumed to be in such a direction that its projections make angles of 
45° with the ground line (16). For the direction of the light it- 
self, corresponding to these projections of it, see PL IX., Fig. 80. 

Let a cube be placed so that one of its faces, L'L^, shall coin- 
cide with a vertical plane, and another face, h"ah lf with the 
horizontal plane. . The diagonals, L'L, and L"L 1? of these faces, 
will be the projections of a ray of light, and the diagonal, LL l7 of 
the cube will be the ray itself ; for the point of which L' and I/' 
are the projections must be in each of the projecting perpendicu- 
lars L'L' and L"L ; hence at L, their intersection. Now the 
right-angled triangles, LL"Lx and LL'L 1? are equal, but not isos- 
celes ; hence the angles LL X L" and LLiL', which the ray itself, 
LL X , makes with the planes of projection are equal, but less than 
45°. Again: in the triangle LLiGr, the angle LLjG- is that made 
by the ray LL X with the ground line, and is more than 45°. 

165. Having now stated the elementary facts and principles 
concerning shadows themselves, we proceed to show how to find 
their projections, by which they are represented 

By (162) we have first to learn how to find where a given line, 
as a ray of light, pierces a given surface. It will be sufficient 
here to show how to construct the points in which a line pierces 
the planes of projection. 

Preliminary Problems. 

Problem 1st. To find where a line drawn through a given 
point, pierces the horizontal plane. 

The construction is shown pictorially in Fig. 1, and in actual 
projection in Fig. 2. 

Let A, Fig. 1, be the given point, whose projections are repre- 
sented by a and a'. If a line passes through a point, its projec- 
tions will pass through the projections of that point; therefore 
if AB be the line, ab and a'V will be its projections. 

Now, first, the required point, C, being in the horizontal plane, 



SHADOWS. 



69 



its vertical projection, c', will be on the ground line; second, the 
same point being a point of the given line, its vertical projection 
will be on the vertical projection, a'b'c', of the line, hence at the 
intersection, c', of that projection with the ground line; third, the 
horizontal projection, c, of the point, which is also the point itself, 
C, will be where the perpendicular to the ground line from c' meets 
abc, the horizontal projection of the line. 

This explanation may be condensed into the following rule. 

1st. Note where the vertical projection of the given line, pro 
duced if necessary, meets the ground line. 

2d. Project this point into the horizontal projection of the line; 
which will give the required point. 

Fig. 2, shows the application of this rule in actual projection. 
ah — a'U is the given line, and by making the construction, we find 
c, where it pierces the horizontal plane. 




Fig. 1. 



Example. — If b d (both figures) had been less than b'd, the line 
would have pierced the horizontal plane behind the ground line, 
that is in the horizontal plane produced backwards. Let this con- 
struction be made, both pictorally, and in projection. 

Problem Id. — To find where a line, drawn through a given pointy 
■pierces the vertical plane. 

See Figs. 3 and 4. The explanation is entirely similar to the 
preceding. The required point, c', being somewhere in the vertical 
plane, its horizontal projection, c, will be on the ground line. It 
being also on the given line, its horizontal projection is on the 
horizontal projection, abc, of the line. Hence, at once, che rule. 



:o 



SHADOWS. 



1st. Xote where the horizontal projection of the given line meeta 
the ground line. 




Fia.8. 



Fig. 4 



2d. Project this point into the vertical projection of the line. 
which will give if*=C, the vertical projection of the point, which is 
also the [ nnf itself 

Example. — Here likewise, if b'd be less than Id* the line will 
pierce the vei the ground line, that is in the vi 

cal plane extended downward. Let the construction be ma'h, 

166. Thr r: : :.-d line is. really, the of the verti- 
cal plane; also the I trace of the horizontal plane. Hence 
the traces : any other horizontal or vertical planes may be called 
the ground lines of those pk ad therefore the preceding con- 
structions may be applied to finding the shadows on such planes. 
as will shortly be seen in the solution of problems. 

167. In res ect to the remaining principles of (162) it only re- 
mains to note that, as two po:n:s determine a straight line, it is 
sufficient to find the sh low :: twe : ". of a straight line, when 

shadow falls on a plane. But if the of the shadow is 

known as in (162 — - : . ~' : . or if we know where the line meets a 
plane surface receiving the shadow, (162 — 5th) it will be sufficient 
to construct one point of the shadow. 

§ II. — P ■;: . ; '. .:." Problems. 

168. Pp.ob. 1. To find, the shadow of a vertical beam, upon a 
vertical -'II K. IX.. Fisr. SI. Let AA' be the beam, let the 



SHADOWS. 71 

vertical plane of projection be taken as the vertical wall, and let the 
light be indicated by the lines, as ab — a"b f . The edges which cast 
the visible shadow are a — a! a" ; ac — a" ; and ce — a'V. The sha- 
dow of a — a'a" is a vertical line from the point 5', which point is 
where the ray from a — a" pierces the vertical plane, ab — a"b' 
pierces the vertical plane in a point whose horizontal projection is 
b. V must be in a perpendicular to the ground line from b (Art. 15), 
and also in the vertical projection, a"b\ of the ray, hence at b'. b'b 
is therefore the shadow of a— a" a'. The shadow of ac — a" is the 
line b'd\ limited by d', the shadow of the point c — a" (162). The 
shadow of ce — a"e' begins at d\ and is parallel to ce — a'V, but is 
partly hidden. 

169. Execution. — The boundary of a shadow being determined, 
its surface is, in practice, indicated by shading, either with a tint of 
Indian ink, or by parallel shade lines. The latter method, affording 
useful pen practice, may be profitably adopted. 

170. Prob. 2. To find the shadow of an oblique timber, which is 
parallel to the vertical plane, upon a similar timber resting against 
the back of it. PL IX., Fig. 82. Let AA' be the timber which 
casts a shadow on BB', which slants in an opposite direction. The 
edge ac — a'c' of AA', casts a shadow, parallel to itself, on the front 
face of BB' ; hence but one point of this shadow need be con- 
structed. Two, however, are found, one being a check upon the 
other. Any point, aa', taken at pleasure in the edge ac — a'c' casts 
a point of shadow on the front plane of BB', whose horizontal 
projection is £, and whose vertical projection (see Prob. 1) must be 
in the ray a'b' ', and in a perpendicular to the ground line at b/ 
hence it is at b' '. The shadow of ac — a'c' being parallel to that 
line, b'd' is the line of shadow, d', the shadow of c — c 7 , was found 
in a similar manner to that just described. 

It makes no difference that b f is not on the actual timber, BB'; 
for the face of that timber is but a limited physical plane, forming 
a portion of the indefinite immaterial plane, in which bb' is found ; 
hence the point V is as good for finding the direction of the indefi- 
nite line of shadow, b'd', as is d', on the timber BB', for finding the 
real portion only of the line of shadow, viz. the part which lies 
across BB'. Here, bd is taken as a ground line (166). 

Observe that the shade line Ab, of the timber AA' should end 
at bb f , where the timbers intersect. 

Example. — Make the timbers larger, A A' more nearly hori- 
zontal than vertical, and then, in both figures, find the shadow 
on the plan. 



72 SHADOWS. 

171. Peob* 3. To find the shadow of a fragment of a horizon- 
tal timber, upon the horizontal top of an abutment on which the 
timber rests. PI. IX., Fig. 83. Let AA'be the timber, and BB' 
the abutment. The vertical edge, a — a"a\ casts a shadow, ab, in 
the direction of the horizontal projection of a ray (162, ±th), and 
limited by the shadow of the point a — a\ The shadow of a — a f is 
at bb', where the ray ab — a'b' pierces the top of the abutment ; b* 
being evidently the vertical projection of this point, and b being 
both in a perpendicular, b'b, to the ground line and in ab, the hori- 
zontal projection of the ray. The shadow of ad — a' is bb", parallel 
to ad — a', and limited by the ray a'b' — db". The shadow of 
de — a'e' is b"c, parallel to de — a'e', and limited by the edge of the 
abutment (162, 2d). Here, a"b' is used as a ground line (166). 

172. Peob. 4. To find the shadow of an oblique timber, upon a 
horizontal timber into which it is framed. PI. IX., Fig. 84. 
The upper back edge, ca — c'a f , and the lower front edge through 
ee', of the oblique piece, are those which cast shadows. By con- 
sidering the point, c, in the shadow of be, Fig. 78, it appears that 
the shadow of ac — aV, Fig. 84, begins at cc', where that edge 
pierces the upper surface of the timber, BB', which receives the 
shadow. Any other point, as aa', casts a shadow, bb', on the plane 
of the upper surface of BB', whose vertical projection is evidently 
b', the intersection of the vertical projection, a'b', of the ray ab — a'b' 
and the vertical projection, e'e', of the upper surface of BB', and whose 
horizontal projection, b, must be in a projecting line, b'b, and in the 
horizontal projection, ab, of the ray. Likewise the line through ee', 
and parallel to cb, is the shadow of the lower front edge of the 
oblique timber upon the top of BB'. This shadow is real, only so 
far as it is actually on the top surface of BB', and is visible and 
therefore shaded, only where not hidden by the oblique piece. ^ 
Where thus hidden, its boundary is dotted, as shown at ea.. The 
point, bb', is in tha plane of the top surface of BB', produced. 

Remark. — Thus it appears that when a line is oblique to a plane 
containing its shadow, the direction of the shadow is unknown till 
found. Let this and the following figures be made much larger. 

Examples. — 1st. Find the shadow when the oblique timber is 
more nearly vertical than horizontal. 

2d. Let the oblique timber ascend to the right. 

173. Peob. 5. To find the shadow of the side wall of a -flight of 
Bteps upon the faces of the steps. PI. IX., Fig. 85. The steps 
can be easily constructed in good proportion, without measure 



SHADOWS. 73 

ments, by making the height of each step two-thirds of its width, 
taking four steps, and making the piers rectangular prisms. 

The edges, aa" — a' and a — ra', of the left hand side wall are 
those which cast shadows on the steps. 

The former line casts horizontal shadows, as b b" — b', parallel to 
itself, on the tops of the steps (162, 2cZ), and shadows, as be" — b'c', 
on the fronts of the steps, in the direction of the vertical projection, 
a'd', of a ray of light (162, 4th) — from the upper step down to the 
shadow of the point aa' '. Likewise, the edge a — ra' casts vertical 
shadows, as g — g'h', on the fronts of the steps (161), and shadows 
on their tops, parallel to ad, the horizontal projection of a ray 
(162, 4th) from the lower step, up to dd', the shadow of aa'; which 
is therefore, where the shadows of aa" — a', and a — ra', meet, a'd' 
is the vertical projection of all rays through points of aa" — a'; 
hence project down b', etc., to find the parallel shadows, b"b, etc. 
Likewise project up g, etc., to find the shadows, g'h', etc., ad being 
the horizontal projection of all rays through points of a — ra' . 

Examples. — 1st. Yary the proportions of the steps and the direo 
Hon of the light ; and in each case, find the shadows, as above. 

2d. By preliminary problems 1st and 2d, find directly where 
the ray through aa' pierces the steps, only remembering that the 
projections, d and d', must be on the same surface. 

3d. Let the piers be eut off by a plane parallel to that of the 
front edges of the steps. (Use an end elevation.) 

174. Prob. 6. To find the shadow of a short cylinder, or washer, 
upon the vertical face of a board. PL IX.. Fig. 86. Since the 
circular face of the washer is parallel to the vertical face of the 
board, BB', its shadow will be an equal circle (161, 3d), of which 
we have only to find the centre, 00'. This point will be the sha- 
dow of the point CC of the washer, and is where the ray CO — CO' 
pierces the board BB'. The elements, rv — r' and tu — t' , which 
separate the light and shade of the cylindrical surface, have the 
tangents r'r" and t't" for their shadows. These tangents, with 
the semicircle t"r", make the complete outline of the required 
shadow. 

175. Prob. 7. To find the shadow of a nut, upon a vertical 
surface, the nut having any position. PL IX., Fig. 86. Let 
a'c'e' — ace be the projections of the nut, and BB' the projections 
of the surface receiving the shadow. . The edges, a'c' — ac and 
ce — e'e', of the nut cast shadows parallel to themselves, since they 
are parallel to the surface which receives the shadow, a'n' — an 



74 SHADOWS. 

are the two projections of the ray which determines the joint of 
shadow, nn' / c'o' — co are the projections of the ray used in find- 
ing 00', and e'r' — er is the ray which gives the point of shadow, 
rr'. The edges at aa' and ee', which are perpendicular to BB', cast 
shadows aV and e'r', in the direction of the projection of a ray of* 
light on BB'. (See cb', the shadow of cb, PL IX., Fig. 78.) 

176. Prob. 8. To find the shadow of a vertical cylinder, on a 
vertical plane. PL IX., Fig. 87. The lines of the cylinder, CC, 
which cast visible shadows, are the element a — a" a', to which the 
rays of light are tangent, and a part of the upper base. The 
shadow of a — a," a', is gg', found by the method given in Prob. 1. 
At g', the curved shadow of the upper base begins. This is found 
by means of the shadows of several points, bd', cc', dd', &c. Each 
of these points of shadow is found as g was, and then they are 
connected by hand, or by the aid of the curved ruler. 

It is well to construct one invisible point, as u', of the shadow, to 
assist in locating more accurately the visible portion of the curved 
shadow line. 

177. Prob. 9. To find the shadow of a horizontal beam, upon 
the slanting face of an oblique abutment. PL IX., Fig. 88. The 
simple facts illustrated by PL IX., Figs. 78-79, have no reference 
to the case of surfaces of shadow, other than vertical or horizontal. 
But they illustrate the fact that the point where a shadow, as aa', 
PL IX., Fig 78, on one surface, meets another surface, is a point 
(a') of the shadow a'b' upon that surface. Thus this problem may 
be solved in an elementary manner by proceeding indirectly, i. e., 
by finding the shadows on the horizontal top of the abutment and 
on its horizontal base. The points where these shadows meet the 
front edges of this top and this base, will be points of the shadow 
on the slanting face, qne. 

By Prob. 3 is found gc, the shadow of the upper back edge, 
ax — aY, of the timber, AA', upon the top of the abutment, c, the 
point where it meets the front edge, ec, is a point of the shadow of 
A A' on the inclined face. By a similar construction with any ray, 
as bp — b'p', is found qp, the shadow of c'b' — db upon the base of 
the abutment; and^, where it intersects nq, is a point of the shadow 
of db — c'b' on the face, qne. The point, dd', where the edge, 
db — c'b', meets dc, is another point of the shadow of that edge; 
hence dq — d'q' is the shadow of the front lower edge, db — c'b', on 
the inclined face of the abutment. The line through cc', parallel to 
dq — d'q\ is the shadow of the upper back edge, ax — a'x', and com- 
p tes the solution. 



SHADOWS. ^5 

178. Prob. 10. To find the shadow of a pair of horizontal tim- 
bers, which are inclined to the vertical plane, upon that plane. PL 
) X., Fig. 89. Let the given bodies be situated as shown in the 
diagram. In the elevation we see the thickness of one timber only, 
because the two timbers are supposed to be of equal thickness and 
halved together. As neither of the pieces is either parallel or 
perpendicular to the vertical plane, we do not know, in advance, 
the direction of their shadows. It will therefore be necessary to 
find the shadows of two points of one edge, and one, of the diago- 
nally opposite edge, of each timber. The edges which cast shadows 
are ac — aV and ht — e'h', of one timber, and ed — e'k and mv — a'm" 
of the other. All this being understood, it will be enough to point 
out the shadows of the required points, without describing their 
construction. (See bb', Fig. 81.) W is the shadow of aa', and dd' 
is the shadow of cc' ; hence the shadow line b f — d is determined. 
So uu' is the shadow of hh' ; hence the shadow line u'w' may be 
drawn parallel to b'd '. Similarly, for the other timber, ff is the 
shadow of ee', and oo f of ram'. One other point is necessary, which 
the student can construct. The process might be shortened some- 
what by finding the shadows of the points of intersection, p andr, 
which would have been the points p" and r" of the intersection of 
the shadows, and thus, points common to both shadows. 

1 79. Prob. 11. To find the shadow of a pair of horizontal tim- 
bers, which intersect as in the last problem, upon the inclined face 
of an oblique abutment. PI. IX., Fig. 90. This problem is so 
similar to Prob. 9, that we only note, as a key to it, that sd, corres- 
ponding to pq, Fig. 88, is the auxiliary shadow of the edge, Ab — n'b' 9 
upon a horizontal plane s'df , cutting the line sb from the face C of 
the abutment, and thence giving a point ss', intersection of sd, 
parallel to Ab, with sb of the shadow of Ab—n'b f on C. 

The following examples of the very useful method of auxiliary 
shadows (162, 6th), and Prob. 9, are here briefly added: 

Examples. — 1st. To find the shadow of an abacus of any form, 
upon a conical column. PI. IX., Fig. 88a. Circles OA and OB, 
with A'B'C represent the conical pillar; and circle OC, with C'D'B', 
its cylindrical cap, or abacus. Then F'Q' is a horizontal plane, 
cutting from the pillar the circle P'Q', of radius OB=wP; and 
pierced by the ray Oa— O'a/ at a'a ; centre of the circle, of radius 
Od=0'D'. Then d, projected at d' , and intersection of the circle 
Od with the circle OQ, is a point of the shadow of circle OC — O'D' 
of the abacus, upon the pillar; since the circle Od is the auxiliary 



76 

' U SHADOWS. 

shadow of the circle OC — O'D' upon the plane P'Q' (162, 3d). Other 
points may be likewise found. 

2d. To find the shadow of the front circle of a niche, upon its 
own spherical surface. PI. IX., Fig. 88b. ABC — A'D'C is the 
quarter sphere, which surmonts the vertical cylindrical part of the 
niche below the line ABC — A'O'C. Then the ray Oa — O'a meets 
any plane BS, parallel to the face, AOC — A'D'O', of the niche, at 
aa' / giving circle a / ) m'n'= circle O'A', for the auxiliary shadow of 
circle 0'A / upon the plane BS (162, 3d), circle a , m'n then cuts 
circle BS — B'cTS', cut from the spherical part of the niche by the 
plane BS, at d ' d 1 ', a required point of shadow. Find other points 
likewise, and join them. The tangent ray at t' gives that point. 

3d. To find the shadoio of a verticcd staff, upon a hemispherical 
dome. PL IX., Fig. 88c. Circle OC, with C'P'D' is the hemi- 
sphere, and A— A'B' the staff. By (162, Uh) Ae the horizontal 
projection of a ray, is also the shadow of A — A'B' upon the assumed 
horizontal plane P'Q', and cutting from the hemisphere the circle 
P Q' — PQ. Then dd! ', intersection of Ae with circle PQ— P'Q ', is 
one point of the required shadow of A— A'B' upon the dome. 

180. Peob. 12. To find the shadoio of the floor of a bridge 
upon a verticcd cylindrical abutment. PI. IX., Fig. 91. The 
line ag — a'g' is the edge of the floor which casts the shadow. 
bdg—b'e'g'n is the concave vertical abutment receiving the shadow. 
gg', where the edgeag — g'a' of the floor meets this curved wall, is 
one point of the shadow, fis the horizontal projection of the point 
where the ray, ef — e'f, meets the abutment ; its vertical projection 
is in e'f, the vertical projection of the ray, and in a perpendicular 
to the ground line, through f hence at/'. Similarly we find the 
points of shadow, dd' and bb', and joining them with /'and g', have 
the boundary of the required shadow. Observe, that to find the 
shadow on any particular vertical line, as b — b"b', we draw, the ray 
in the direction b — a ; then project a at a', &c. 

Remark. — The student may profitably exercise himself in chang- 
ing the positions of the given parts, while retaining the methods 
of solution now given. 

For example, let the parts of the last problem be placed side 
by side, as two elevations, giving the shadow of a vertical wall 
on a horizontal concave cylindrical surface; or, let the timbers, Fig. 
89, be in vertical planes, and let their shadows then be found on a 
horizontal surface. 




"lit * 1:1 " i W 



1 I oi/ij l "^^ ^Hr — ^ — f — ^ 



PL. IX 




CHAPTER II. 

SHADING. 

131. The distinction between a shade and a shadoiv is this. 
A shadoiv, as indicated by the preceding problems, is the portion 
of a body from which light is excluded by some other body. A 
shade, is that portion of the surface of a body from which light 
is excluded by the body itself (158, 159). 

The accurate representation of shades assists in judging of the 
forms of bodies ; that of shadows is similarly useful, besides aid- 
ing in showing their relations io surrounding bodies. 

In either case, a flat tint mainly shows only where the shade, 
or shadow, is ; while finished shading helps to show the form and 
position of the body. 

182. Example 1°. To shade the elevation of a vertical right hex- 
agonal prism, and its shadow on the horizontal plane. PL X., Figs. 
02 and A. Let the prism be placed as represented, at some dis- 
tance from the vertical plane, and with none of its vertical faces 
parallel to the vertical plane. The face, A, of the prism is in the 
light ; in fact, the light strikes it nearly perpendicularly, as may be 
seen by reference to the plan ; hence it should receive a very light 
tint of indian ink. The left hand portion of the face, A, is made 
slightly darker than the right hand part, it being more distant; 
for the reflected rays, which reach us from the left hand portion, 
have to traverse a greater extent of air than those from the neigh- 
borhood of the line tt\ and hence are more absorbed or retarded ; 
since the atmosphere is not perfectly transparent. That is, these 
rays make a weaker impression on the eye, causing the left hand 
portion of A, from which they come, to appear darker than at tt' 

Remarks. — a. It should be remembered that the whole of 
face A is very light, and the difference in tint between its opposite 
sides very slight. 

b. As corollaries from the preceding, it appears: first, that a sur- 
face parallel to the vertical plane would receive a uniform tint 
throughout ; and, second, that of a series of such surfaces, all of 



SHADING 



which are in the light, the one nearest the eye would be lightest, 
and the one furthest from the eye, darkest. 

c. It is only for great differences in distance that the above effects 
are manifest in nature ; but drawing by projections being artificial, 
both in respect to the shapes which it gives in the drawings, and in 
the absence of surrounding objects which it allows, we are obliged 
to exaggerate natural appearances in some respects, in order to 
convey a clearer idea of the forms of bodies. 

d. The mere manual process of shading small surfaces is here 
briefly described. With a sharp-pointed camel' s-hair brush, wet 
with a very light tint of indian ink, make a narrow strip against the 
left hand line of A, and soften off its edge with another brush 
slightly wet with clear water. When all this is dry, commence at 
the same line, and make a similar but wider strip, and so proceed 
till the whole of face A is completed, when any little irregularities' 
in the gradation of shade can be evened up with a delicate sable 
brush, damp only with very light ink. 

183. Passing now to face B, it is, as a whole, a little darker than 
A, because, as may be seen by reference to the plan, while a beam 
of rays of the thickness np strikes face A, a beam having only a thick 
ness pv, strikes lace B ; i. e., we assume, first, that the actual bright- 
ness of a flat surface is proportioned to the number of rays of light 
which it receives; and, second, that its apparent brightness is, other 
things being the same, proportioned to its actual brightness. Also, 
the part at a — a' being a little more remote than the line t — t', the 
part at a — a' is made a very little darker. 

184. The face C is very dark, as it receives no light, except 
a small amount by reflection from surrounding objects. This 
side, C, is darkest at the edge a — a' which is nearest to the 
eye. This agrees with experience; for while the shady side 
of a house near to us appears in strong contrast with the 
illuminated side, the shady side of a remote building appears 
scarcely darker than the illuminated side. This fact is explained 
as follows: The air, and particles floating in it, between the 
eye and the dark surface, C, are in the light, and reflect some 
light in a direction from the dark surface C to the eye; and 
as the air is invisible, these reflections appear to come from 
that dark surface. Now the more distant that surface is, the 
greater will be the body of illuminated air between it and the 
eye, and therefore the greater will be the amount of light, appa- 
rently proceeding from the surface, and its consequent apparent 
brightness. That is, the more distant a surface in the dark is, the 



SHADING. ?9 

lighter it will appear. It may be objected that this would make ont 
the remoter parts of illuminated surfaces as the lighter parts. But 
not so ; for. the air is a nearly perfect transparent medium, and hence 
reflects but little, compared with what it transmits to the opaque 
body ; but being not quite transparent, it absorbs the reflected rays 
from the distant body, in proportion to the distance of that body, 
making therefore the remoter portions darker ; while the very weak 
reflections from the shady side are reinforced, or replaced, by more 
of the comparatively stronger atmospheric reflections, in case of the 
remote, than in case of the near part of that shady side. Thus is 
made out a consistent theory. 

In relation, now, to the shadow, it will be lightest where furthest 
from the prism, since the atmospheric reflections evidently have to 
traverse a less depth of darkened air in the vicinity of de — d' than 
near the lower base of the prism at abc. 

185. Ex. 2°. To shade the elevation of 'a vertical cylinder. PI. X., 
Fig. 93. Let the cylinder stand on the horizontal plane. The figures 
on the elevation suggest the comparative depth of color between 
the lines adjacent to the figures. The reasons for so distributing 
the tints will now be given. See also Fig. B. 

The darkest part of the figure may properly be assumed to be 
that to which the rays of light are tangent ; viz., the vertical line at 
tt\ from which the tint becomes lighter in both directions. 

The lightest line is that which reflects most light to the eye. Now 
it is a principle of optics that the incident and the reflected ray 
make equal angles with a perpendicular to a surface. But nC is the 
incident ray to the centre, and Ce the reflected ray from C to the 
eye (12). Hence d, which bisects ne, shows where the incident ray, 
ed, and reflected ray, dq, make equal angles with the perpendicular 
(normal) dC, to the surface. Hence d — d' is the lightest element. 

186. Remark. — The question may here arise, "If all the light 
that is reflected towards the eye is reflected from d — as it appears 
to be — how can any other point of the body be seen ?" To answer 
this question requires a notice— -first, of the difference between 
polished and dull surfaces ; and, second, between the case of light 
coming wholly in one direction, or principally in one direction. If 
the cylinder CC, considered as perfectly polished, were deprived of 
all reflected light from the air and surrounding objects, the line at 
d — d' would reflect to the eye all the light that the body would 
remit towards the eye, and would appear as a line of brilliant light, 
while other parts, remitting no light whatever, would be totally 



SO SHADING. 

invisible. Let us now suppose a reflecting medium, though an 
imperfect one, as the atmosphere, to be thrown around the body. 
By reflection, every part of the body would receive some light from 
all directions, and so would remit some light to the eye. making the 
body visible, though faintly so. But no body has a polish that is 
absolutely perfect; rather, the great majority of those met with iD 
engineering art have entirely dull surfaces. Xow a dull surface, 
greatly magnified, may be supposed to have a structure like that 
shown in PL X. 5 Fig. 97, in which many of the asperities may be 
supposed to have one little facet each, so situated as to remit to the 
eye a ray received by the body directly from the principal source 
of light." 

187. Having thus shown how any object placed before our eyes 
cao be seen, we may proceed with an explanation of the distribu- 
tion of tints, b is midway between d and t. At b, the ink may be 
diluted, and at e — e\ much more diluted, as the gradation from a 
faint tint at e to absolute whiteness at d should be without any 
abrupt transition anywhere. 

The beam of incident rays which falls on the segment dn, is 
broader than that which strikes the equal segment de / hence the 
segment nd is, on the elevation, marked 5, as being the lightest band 
which is tinted at all. The segment nr, being a little more obliqueh 
illuminated, is less bright, and in elevation is marked 3, and may be 
made darkest at the left hand limit. Finally, the segment rv receives 
about as much light as rn, but reflects it within the very narrow 
limits, s, hence appears brighter. This condensed beam, s, of 
reflected rays would make rv the lightest band on the cylinder, but 
for two reasons; first, on account of the exaggerated effect allowed 
to increase of distance from the eye ; and, second, because some of 
the asperities, Fig. 97, would obscure some of the reflected rays 
from asperities still more remote; hence rv is, in elevation, marked 
4, and should be darkest at its right hand limit. 

188. The process of shading is the same as in the last exercise. 
Each stripe of the prelinrinary process may extend past the preced- 
ing one, a distance equal to that indicated by the short dashes at 
the top of Fig. 94. When the whole is finished, there should be a 
uniform gradation of shade from the darkest to the lightest line, 
free from all sudden transitions and minor irregularities. 

189. Ex. 3°. To shade a right cone standing upon the horizon- 
tal plane, together with its shadow. PL X., Fig. 95. The sha- 
dow of the cone on the horizontal plane, will evidently be bounded 



SHADIJNG. 



81 



by the shadows of those lines of the convex surface, at which the 
light is tangent. The vertex is common to both these lines, and 
casts a shadow, v'". The shadows being cast by straight lines of 
the conic surface, are straight, and their extremities must be in the 
base, being cast by lines of the cone, which meet the horizontal 
plane in the cone's base; hence the tangents v'"t and v'"t" are the 
boundaries of the cone's shadow on the horizontal plane, and the 
lines joining t' and t" with the vertex are the lines to which th< 
rays of light are tangent ; i. e., they are the darkest lines of the 
shading ; hence tv, the visible one in elevation, is to be vertically 
projected at t'v' . 

The lightest line passes from vv' to the middle point, y, between 
n and p in the base. At q and at^t? a change in the darkness of the 
tint is made, as indicated by the tigures seen in the elevation. In 
the ease of the cone, it will be observed that the various bands of 
color are triangular rather than rectangular, as in the cylinder ; so 
that great care must be taken to avoid filling up the whole of the 
upper part of the elevation with a dark shade. See PL X., Fig. C. 

190. Ex. 4°. To shade the elevation of a sphere. PL X., Fig. 
96. It is evident that, if there be a system of parallel rays, tan- 
gent to a sphere, their points of contact will form a great circle, 
perpendicular to these tangents; and which will divide the light 
from the shade of the sphere. That is, it will be its circle of 
shade. Each point of this circle is thus the point of contact of one 
tangent ray of light. If now, parallel planes of rays, that is, planes 
parallel to the light, be passed through the sphere, each of them 
will cut a circle, great or small, from the sphere, and there will be 
two rays tangent to it on opposite sides, whose points of contact 
will be points of the circle of shade. 

In the construction, these parallel planes of rays will be taken 
perpendicular to the vertical plane of projection. 

Next, let us recollect that always, when a line is parallel to a plane, 
its projection on that plane will be seen in its true direction. Now 
BD' being the direction of the light, as seen in elevation, let BD' be 
the trace, on the vertical plane of projection — taken through the 
centre of the sphere — of a new plane perpendicular to the vertical 
plane, and therefore parallel to the rays of light. The projection 
of a ray of light on this plane, BD', will be parallel to the ray itself, 
and therefore the angle made by this projection with the trace BD' 
will be equal to the angle made by the ray with the vertical plane. 
But, referring to PI. IX., Fig. 80, we see that in the triangle 



82 SHADING. 

LLXj, containing th 3 angle LL,L' made by the ray LL t with the ver- 
tical plane, the side L'Li is the hypothenuse of the triangle ~L'b L„ 
each of whose other sides is equal to LI/. Hence in PI. IX., 
Pig. 96, take any distance, Be, make AB perpendicular to BD', and 
On it lay off BD = Bc, then make BD' = Dc, join D and D', and DD' 
will be the projection of a ray upon the plane BD', and BD'D will 
be the true size of the angle made by the light with the vertical 
plane; it being understood that the plane BD', though in space 
perpendicular to the vertical plane, is, in the figure, represented as 
revolved over towards the right till it coincides with the vertical 
plane of projection, and with the paper. 

191. We are now ready to find points in the curve of shade, oo' 
is the vertical projection of a small circle parallel to the plane BD' y 
and also of its tangent rays. The circle og'o\ on oo' as a diameter, 
represents the same circle revolved about oo' as an axis and into 
the vertical plane of projection. Drawing a tangent to og'o\ 
parallel to DD', we find g\ a visible point of the curve of shade, 
which, when the circle revolves back to the position oo\ appears at 
{/, since, as the axis oo' is in the vertical plane, an arc, g'g, described 
about that axis, must be vertically projected as a straight line. (See 
Art. 31.) 

In a precisely similar manner are found A, &, m, and f. At A 
and B, rays are also evidently tangent to the sphere. Through 
A, f, &c., to B the visible portion of the curve of shade may now 
be sketched. 

192. The most highly illuminated point is 90° distant from the 
great circle of shade ; hence, on ABQ, the revolved position of a 
great circle which is perpendicular to the circle of shade, lay off 
^'Q = g-B=the chord of 90°, and revolve this perpendicular circle 
back to the position qq\ when Q will be found at r'. But the 
brilliant point, as it appears to the eye, is not the one which receives 
most light, but the one that reflects most, and this point is midway, 
in space, between r' and r, i. e. at P, found by bisecting QB, and 
drawing EP; for at K the incident ray whose revolved position is 
parallel to Qr or DD', and the reflected ray whose revolved posi- 
tion is parallel to rB, make equal angles with R'K, the perpen- 
dicular (normal) to the surface of the sphere. See PI. X., Fig. D. 

193. In regard now to the second general division of the problem 
— the distribution of tints ; a small oval space around P should be 
left blank. The first stripe of dark tint reaches from A to B, along 
the curve of bhade, and the successive stripes of the same tint extend 
to BqA on one side of B&A, and to oeu) on the other. Then take 



SHADING. 83 

a lighter tint on the lower half of the next zone, and a still 
lighter one on its upper half (2) and (3). In shading the next 
zone, use an intermediate tint (3-4), and in the zone next to P a 
yery light tint on the lower side (4), and the lightest of all on 
the upper side (5). After laying on these preliminary tints, 
even up all sudden transitions and minor irregularities as in 
other cases. 

194. Ex. 5°. Shades and Shadows on a Model. PL 
XL, Figs. 1 and 2. General Description. — This plate contains 
two elevations of an architectural Model. It is introduced as 
affording excellent practice in tinting and shading large surfaces, 
and useful elementary studies of shadows. The construction of 
these elevations from given measurements is so simple, that only 
the base and several centre lines need be pointed out. 

QR is the ground line. ST is a centre line for the flat topped 
tower in Fig. 1. UV is a centre line for the whole of Pig. 2, ex- 
cept the left-hand tower and its pedestal. WX is a centre line 
for the tower through which it passes. YZ is the centre line for 
the roofed tower in Fig. .1. The measurements are recorded in 
full, referred to the centre lines, base line, and bases of the towers, 
which are the parts to be first drawn. 
Graphical construction of the shadows. 

1°. The roof, D — DT>", casts a shadow on its tower. The point, 
EE', casts a shadow where the ray, Ee, pierces the side of the tower. 
e is one projection of this point ; e\ the other projection of the same 
point, is at the intersection of the line ee"e' with the other projection, 
EV, of the ray. The shadow of a line on a parallel plane (162) is 
parallel to itself, hence e'f, parallel to E'F', is the shadow of E'F'. 

The shadow of DE — D'E' joins e' with the shadow of D — D'. 
The point d, determined by the ray Dc7, is one projection of the 
latter shadow ; the other projection, d\ is at the intersection of 
dd"d' with the other projection, D'cf , of the ray. d' is on the side 
of the tower, produced, hence e'd' is only a real shadow line from 
e' till it intersects the edge of the tower. 

Remembering that the direction of the light is supposed to 
change with each position of the observer, so that as he faces each 
side of the model, in succession, the light comes from left to right 
and from behind his left shoulder, it appears that the point, DD", 
casts a shadow on the face of the tower, seen in Fig. 2, and that 
D"d"" will be the position of the ray, through this point, on Fig. 



84 SHADING. 

1. The point d fn is therefore one projection of the shadow of 
DD". The other is at d'"\ the intersection of the lines d'"d"" with 
T>d"'\ the other projection of the ray. Likewise EE" casts a 
shadow, e"'e'"\ on the same face of the tower, produced. DD"', 
being parallel to the face of the tower now being considered, its 
shadow, d""q, is parallel to it. The line from d"" towards e"", 
till the edge of the tower, is the real portion of the shadow of 
DE— D"E". 

From the foregoing it will be seen how most of these sha- 
dows are found, so that each step in the process of finding similar 
hadows will not be repeated. 

2°. The body of the building — or model — casts a shadow on the 
roofed tower, beginning at AA' (161). The shadow of BB' on the 
Bide of this tower is bb', found as in previous cases, and A'b' is the 
shadow of AB — A'B'. From b' downwards, a vertical line is the 
Bhadow of the vertical corner edge of the body of the model upon 
the parallel face of the tower. 

3°. The line CC" — C, which is perpendicular to the side of the 
roofed tower, casts a shadow, C'c', in the direction of the projection 
of a ray of light on the side of the tower. 

4°. In Fig. 1, a similar shadow, s'f, is cast by the edge s' — ss' 
of the smaller pedestal. 

5°. In Fig. 2, is visible the curved shadow, c"rg, cast by the 
vertical edge, at c", of the tower, on the curved part of the pedestal 
of the tower. The point g is found by drawing a ray, G'C — Gg, 
which meets the upper edge of the pedestal at gC. The point c\ 
the intersection of the edge of the tower with the curved part of 
the pedestal, is another point. Any intermediate point, as r, is 
found by drawing the ray RV'y r' is then one projection of the 
shadow of R'R, and the other is at the intersection of the line r'r 
with the other projection Rr of the ray. These are all the shadows 
which are very near to the objects casting them. 

6°. The flat topped tower casts a shadow on the roof of the body. 
The upper back corner, HH', casts a shadow on the roof, of which 
h is one projection and h the other. The back upper edge H — HT 
being parallel to the roof, the short shadow tih", leaving the roof at 
h", is parallel to HT'. The left hand back edge HJ — H'J' casts a 
shadow on the roof, of which hh' is one point. The point JJ' casts 
a shadow jf on the roof produced, h'j is therefore a real shadow 
only till it leaves the actual roof at u. 

7°. The same tower casts a shadow on the vertical side of the 
body, of which j"j"\ found as in previous cases, and w, are points. 



, / 



SHADING. 



85 



the upper back point, KK', of the shaft of the tower, casts a sha- 
dow, kk' , which is joined with /'", giving the shadow of 
JK — J'K'. From W y k'l is the vertical shadow line of the left- 
hand back edge of the flat-topped tower on the parallel plane 
of the side of the body of the model. 

8°. The same tower casts a shadow on the curved — cylindrical 
part of the pedestal. To find the point m' , oi shadow, draw a 
ray, MO, Fig. 2, intersecting the upper edge o ; the pedestal at 
C, which is therefore one projection of the shalow of the point 
MM'. The other projection, m' , of the same thadow, is at the 
intersection of the other projection, Wm' s of he ray, with the 
other projection, m'C, of the edge of the pedetal. The point 
of shadow, nn f , cast by the point NN"', is similary found, and so 
is the point oo', cast by the point 00' of the Tout right-hand 
edge of the tower. Make m'm" — n'o', and fhd intermediate 
points, v'v" , as rr' was found, and the curved shadow on the 
cylindrical part of the pedestal will then be fourd. 

9°. From n' and o' , vertical lines are the shalows of opposite 
diagonal edges of the tower, on the vertical fice of the main 
pedestal. 

10°. This flat-topped tower also casts a shadov on the side of 
the roofed tower. The right back corner, H — I', )f the op, casts 
the shadow li'"li"" on the side of the roofed "ower J through 
which the shadow line, li""x is drawn, paralel to the line 
H — H'l' which casts it. The right-hand top line, I' — H, being 
perpendicular to the plane of the sides of this tower, 3asts the 
shadow h""i' upon it, parallel to the projection of a ra of light. 
(162.) This shadow line is real, only till it leaves the tower at 
z; — i' being in the plane of the side of the tower product. — and it 
completes all the shadows visible in the two elevations 



Errors in Shading, Relative darkness of the light and 



lade, etc. 



195. The most frequent faults to guard against aij — l^zf. A 
brush too wet, or too long applied to one part of the figui ; giving 
a ragged or spotty appearance. 

2d. Outlines inked in black, whereas the form and jtlines of 
actual objects are indicated, not by black edges, but bi contrast 
of shade only, with edges lighter than other parts. 



3d. 



Much too little contrast between the shading of t 



parts in 



86 



SHADING. 



the light and those in the dark. The former should generally, 
as on a cylinder, be very much lighter, or not one quarter as 
dark as the parts in the dark. This is confirmed by examining 
photographs of objects illuminated in the manner supposed in 
this chapter. 

196. There aiefive methods of shading, as follows : 

1st. Softened wet shading. This is done with a large brush, 
quite wet, and iJ applicable to large figures. 

2d. Softened try shading. This is done with a brush, nearly 
dry, and is appljcable to figures of the size of those on PL X., 
or not much larger. 

3d. Shading ty superposed flat tints. This method is very neat 
and effective f oiiarge figures, not to be closely examined. It con- 
sists of the prdminary stripes of the 2d method evenly and 
smoothly done, mthout softening their edges (183 d). 

tth. Stippli'.g, or dot shading. This is done with a fine pen 
as in making saLd in topographical drawing. 

5th. Line shtding. This is done by means of lines of graded 
size and distance apart ; as in wood and lithographic mechanical 
engravings. 

The letails If these methods are further explained in my 
"Drafting Instruments aud Operations." 



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DIVISION FOURTH. 

ISOMETRICAL AND OBLIQUE PROJECTIONS. j 



CHAPTER I. 

FIRST PRINCIPLES OF ISOMETRICAL DRAWING. 

197. It is the object of this Division to explain some methods 
of making drawings, especially of details and various small work, 
which combine the intelligibleness of pictorial figures, with the 
exactness of common projections (Div. I.). Such drawings pos- 
sess, among others, the advantage of being more readily under- 
stood by workmen unacquainted with ordinary projections, than 
plans and elevations might be. 

We shall therefore now explain the methods called Isometri- 
cal and Oblique Projections, taking up the former first. 

198. Isometrical projection is that in which a solid right angle, 
like that at the corner of a cube, is placed so that the three plane 
right angles which bound it appear equal in the projection. 

The elementary principles of this projection are mos't simply 
explained, as follows, by reference to a cube, since this is the 
simplest possible rectangular body. 

For clearness, Ave have to distinguish in a cube, its edges, its 
face diagonals, and its body diagonals. 

199. Accordingly, let PL XII., Fig. 98, represent a cube whose 
front face is parallel to V. This face will therefore be the only 
one visible in elevation, and will appear in its real size. 

Next suppose the cube to be turned horizontally 45°, when 
two of its vertical faces will be shown equally, as in Fig. 99, 
though not in their real size. 

Finally, suppose the cube to be turned up at its back corner 
dd'> about any axis as ef, parallel to the ground line, until the 



88 FIEST PEIXCIPLES OF ISOMETRICAL DRAWING. 

top and the two faces shown in Fig. 99 all appear equal, as in 
Pig. 100. This evidently can be done, for in Fig. 99 the top 
face is not seen at all in vertical projection; but invert the figure, 
taking plan for elevation, and it is fully seen, and the vertical 
faces are unseen; hence there must be one intermediate position, 
where, as in Fig. 100, the three faces seen in Fig. 99 will be 
seen equally. 

200. Results. — 1°. The equality of the projections of the 
three visible faces of the cube, when inclined as in Fig. 100, in- 
cludes the equality of the projections of its edges. Hence (Div. 
I., Art. 11), these edges also being equal in space, are equally 
inclined to the plane of projection. 

2°. The equal projections of the faces include the equal pro- 
jections of their like angles. Hence the three angles at C, and 
the angles equal to them at D, o, p, are angles of 120° each. 
The remaining angles of the projections of the faces are of 60° 
each. 

3°. Lines like Ce", C"D, C"/", Fig. 100, equal, and equally 
inclined to a plane, and proceeding from one point, C, must 
terminate in a plane parallel to the given plane. Hence the 
three face diagonals e['f" , f"n, ne' , are all parallel to the plane 
of projection, and therefore appear in their real size. Hence in 
(199) we might have said, incline the cube, until three visible 
face diagonals, VcV — BD ; ««V— A'B; «'V— A'D, Fig. A, 
become parallel to the plane of projection, XY. All other lines 
of Fig. 100 appear less than their real size. 

4°. By 3°, e"f"n is evidently an equilateral triangle ; and, by 
1° and 2°, all the other lines make equal angles of 30° with 
these. Hence the perimeter Y)f"onpe" is a regular hexagon, 
and the projections of the visible foremost, and invisible hind- 
most corners of the cube coincide at C. Hence the diagonal of 
the cube, which joins these points, is perpendicular to the plane 
of projection. 

201. Definitions. Fig. 100: C is the isometric centre, Ce", 
Cf", Cn, are the isometric axes. Lines parallel to these axes 
are isometric lines ; others are non-isometric lines. Planes par- 
allel to the faces of the cube are isometric plain-*. 

The edges of a cube, or similar rectangular body, arc the lines 



PIRST PRINCIPLES OF ISOMETRICAL DRAWING. 



89 



whose dimensions would naturally be desired. Hence it is usual 
to make Ce", Of, D«", etc., Fig. 100, equal to the edges of the 
cube itself ; as in Fig. 101. Such a figure is, for distinction, 
called the Isometrical Drawing of the cube, and it is the isomet- 
rical pro/ecfaw of an imaginary larger cube, which is to Fig. 101 
as Fig. 99 is to Fig. 100. 

202. Another demonstration* The first principles of iso- 
metrical projection may be developed from the following 

Proposition. The plane which is perpendicular to the body 
diagonal of a cube, is equally inclined to its faces and edges. 

Let abcd-a'b'c'd'a"b"c"d", PL XII., Fig. A, be the plan 
and elevation of a cube, shown as in PI. XII., Fig. 99, only 
that the ground line, GL, is inclined, to permit the construction 
of the isometrical figure in an upright position, as m Fig. 
100, by direct projection from the given elevation. 

The body diagonal, ac— a'c", parallel to V, is the common 
hypothenuse of three equal right-angled triangles, similarly situ- 
ated relative to the cube. One base of each of these triangles is 
an edge, beginning at cc" ; the other is a face diagonal, begin- 
ning at aa' . 

Thus, one of these three triangles is ac — a f c'c", which, being 
parallel to V, shows its real size on V. Another has for its bases 
the edge U—b"c" and the face diagonal ao—a'b"; and the 
third has for its bases the edge dc—d"c", and the face diagonal 

ad—a'd". 

Now since these triangles are thus equal, and similarly placed 
on the cube, their common hypothenuse, the body diagonal 
a'c", making equal angles with the three face diagonals which 
meet at aa', also makes equal angles with the faces containing 
these diagonals. 

Hence a plane of projection XY, or Vi, perpendicular to this 
body diagonal, is equally inclined to the three faces of the cube 
which meet at aa' , or at cc"; which agrees with the enunciation. 

From this conclusion follow all the other particulars in the 
preceding articles. 

In making the figure, H and the plane XY (Yi) are both per- 
pendicular to Y. Hence Cn=cc"; Vo = dd"; etc. 

* This may be omitted at discretion, but may be preferred by Teachers 
and others, as fresher, and more concise and strictly geometrical. 



CHAPTER II. 

PEOBLEMS INVOLVING OXLY ISOMETEIC LIKES. 

207. Peob. 1. To construct the isometrical projections, and 
drawings of cubes, and a rectangular block cut from a corner of 
one of them. 

The principles of the last chapter yield several simple con- 
structions, as follows: 

First Method. Draw an equilateral triangle, as e"f'n, Fig. 
100, each of whose sides shall be equal to a face diagonal of the 
cube. Then (200, 4°) draw two lines, as e"D and e"C, with the 
30° angle of the 30° and 60° triangle, making angles of 30° with 
each side of the triangle e"f"n, at each extremity. These lines 
will intersect as at 0, D, o, etc., forming the isometric projection 
of the cube. By extending Ce" ', Of", On, till equal to the edge 
of the cube, and joining their new outer extremities, the projec- 
tion will be converted into the isometric drawing. 

Second Method. Fig. 100. Draw the isometric axes indefi- 
nitely, Ce" and C/ 7 ' each making angles of 30° with a horizontal 
EF, on which lay off half of a face diagonal ef Fig. 99, each way 
from C, giving E and F. Then perpendiculars at E and F will 
limit the right and left axes at e" and f" . On is then made 
equal to Ce" or C/", and the remaining edges are drawn parallel 
to these three. From this projection, make the draiving as in 
the first method. 

Third Method. Drawing the axes as before, lay off on each 
the true length of an edge of the cube, and complete the figure 
as just described. This at once makes the drawing. 

Fourth Method. Fig. 101. With centre C and radius equal 
to an edge of the cube, draw a circle, and in it inscribe a regu- 
lar hexagon in the position shown, adding the alternate radii 
Ca, Cb, Cc, which again gives the drawing of the cube. 

Let a prismatic block be cut from the front corner of this 
cube. Suppose the edge of the cube to be five inches long, drawn 



PROBLEMS INVOLVING ONLY ISOMETRIC LINES. 91 

to a.scale of one-fifth. Let Ca'=2 inches; Cb'=3 inches; (V= 
1 inch. Lay off these distances upon the axes, and at a', b' , c' , 
draw isometric lines which will be the remaining visible edges of 
the block. 

Examples. — 1st. Draw a square panel in each visible face. 

2d. Draw a square tablet in each visible face. 

3d. Let a cube 1-J- inches on each edge be cut from every visi- 
ble corner of the cube. 

208. Prob. 2. To find the shade lines on a cube, and the 
shadows of isometric lines upon isometric planes. 

By (206), three faces of the cube, PL XII. , Fig. 102, will be 
illuminated by light passing in the direction LL', from a to p. 
Two of these faces, aonC, and aCqD are visible. The under and 
right-hand faces, onpC, npqC, and pq~DG, are in the dark, hence 
by the rule (18) the edges on, nC, Cq and q~D are to be inked 
heavy. 

The shadoios of ab=oa produced ; and of ad— J) a produced. 
ab is perpendicular to aCqD, hence (162, 4:th) its shadow will be 
in the direction of the projection of the light upon that face. 
But aq is evidently the projection of the ray, LL' upon aCqD; 
and as a is where ab meets aCq~D (162, 5th), ab' is the shadow of 
ab on aCqJ). 

In like manner, ad' is the shadow of ad upon aonC. 

Otherwise. A plane containing oa and ap, will contain rays 
through points of oab, and will also contain pq and will therefore 
cut aCqJ) in the line aq. Hence rays from all points of ab will 
pierce aCqD in aq. Hence ab' is the shadow of ab. Again ; 
Dap is the plane of rays, containing Dd and pn, and cutting the 
face aonC in the line an. Hence d' is the shadow of d, and ad' , 
that of ad. 

Similar constructions apply to the isometric projections and 
drawings of all rectangular bodies. 

Examples. — 1st. Find the shadow of the cube on the plane 
of its lower base. 

2d. Find the shadow of ah (considered as Ca produced) upon 
the rear face ~Dao. 

3d. Find the shadow of a line parallel in space to Ca, upon 
the face Caon. 



92 PROBLEMS IXVOLYLXG OXLY ISOMETRIC LliTES. 

tth. Find the shadow of Dq produced, on the plane of" the 
lower base of the cube. 

209. Prob. 3. To construct the isometrical draiving of a car- 
penter's oil-stone box. PL XIII., Fig. 103. This problem involves 
the finding of points which are in the planes of the isometric 
axes. 

Let the box containing the stone be 10 inches long, 4 inches 
wide, and 3 inches high, and let it be drawn on a scale of -J. 

Assume C then, and make Ca=4i inches, CZ>=10 inches, and 
Cc=3|- inches, and by other lines aD, ~Db, &c, equal and parallel 
to these, complete the outline of the box. 

Kepresent the joint between the cover and the box as being 1 
inch below the top, aCb. Do this by making Gp=l inch, and 
through p drawing the isometric lines which represent the 
joint. 

Suppose, now, a piece of ivory 5 inches long and 1 inch wide 
to be inlaid in the longer side of the box. Bisect the lower edge 
at d, make de=l inch, and ee=l inch. Through e and e' , draw 
the top and bottom lines of the ivory, making them 24- inches 
long on each side of de' ', as at e't, e't' . Draw the vertical lines 
at t and i', which will complete the ivory. 

210. To show another way of representing a similar inlaid 
piece, let us suppose one to be in the top of the cover, 5 inches 
long and 1J inches wide. Draw the diagonals ab and CD, and 
through their intersection, o, draw isometric lines; lay off oo'= 
2^- inches, and oo"=^ of an inch, and lay off equal distances in 
the opposite directions on these centre lines. 

Through o', o" , &c, draw isometric lines to complete the 
representation of the inlaid piece. 

PI. XII., Fig. B, suggests other exercises, and illustrates the 
gaining of room by making the longest lines of the figure hori- 
zontal, as is often done in practice, when the longest lines of the 
original object are horizontal, since it is not the position, but 
the form of the figure which makes it isometrical. 

Examples. — 1st. Eeconstruct, isometrically, two courses of 
any of the examples in brick-work on PI. Y. 

2d. Do likewise with any of the figures (40-49) of PI. VII., 
assuming any convenient width for the timber.:- i:i each case 



PL.XII. 




PROBLEMS INVOLVING ONLY ISOMETRIC LINES. 93 

211. Prob. 4. To represent the same box (Prob. 3) with the 
cover removed. PL XIII., Fig. 104. 

This problem involves the finding of the positions of points 
not in the given isometric planes. 

Supposing the edges of the box to be indicated by the same let- 
ters as are seen in Fig. 103, and supposing the body of the box to 
be drawn, 10 inches long, 4 inches wide, and 2 J- inches high; then, 
to find the nearest upper corner of the oil stone, lay off on Cb 1 J 
inches, and through the point /, thus found, draw a line, ff, 
parallel to Go. On ff, lay off 1 inch at each end, and from the 
points hh', thus found, erect perpendiculars, as Jm, each f of an 
inch long. Make the further end, x, of the oil stone l£ 
inches from the further end of the box, and then complete the 
oil stone as shown. To find the panel in the side of the box, lay 
off 2 inches from each end of the box, on its lower edge; at the 
points thus found, erect perpendiculars, of half an inch in length, 
to the lower corners, as p, of the panel; make the panels 1£ 
inches wide, as at pp' , and \ an inch deep, as at pr, which last 
line is parallel to Ca ; and, with the isometric lines through 
r, p, p' ', and t, completes the panel. 

212. TJie manner of sJiading tJiis figure will now be explained. 
The top of the box and stone is lightest. Their ends are a 

trifle darker, since they receive less of the light which is diffused 
through the atmosphere. The shadow of the oil stone on the 
top of the box is much darker than the surfaces just mentioned. 
The shadow of the foremost vertical edge of the stone is found 
in precisely the same way as was the shadow of the wire upon the 
top of the cube, PL XII., Fig. 102. The sides of the oil stone 
and of the box, which are in the dark, are a little darker than the 
shadow, and all the surfaces of the panel are of equal darkness 
and a trifle darker than the other dark surfaces. In order to 
distinguish the separate faces of the panel, when they are of the 
same darkness, leave their edges very light. The little light 
which those edges receive is mostly perpendicular to them, re- 
garding them as rounded and polished by use. These light lines 
are left by tinting each surface of the panel, separately, with a 
small brush, leaving the blank edges, which may, if necessary, 
be afterwards made perfectly straight by inking them with a 



94 PROBLEMS INVOLVING ONLY ISOMETRIC LIKES. 

light tint. The upper and left-hand edges of the panel, and all 
the lines corresponding to those which are heavy in the previous 
figures, may be ruled with a dark tint. In the absence of an 
engraved copy, the figures will indicate tolerably the relative 
darkness of the different surfaces ; 1 being the lightest, and the 
numbers not being consecutive, so that they may assist in denot- 
ing relative differences of tint. When this figure is thus shaded, 
its edges should not be inked with ruled lines in black ink, but 
should be inked with pale ink. 

Examples. — 1st. Eeconstruct, isometrically, any of the figures 
53, 54, 55, 58, on PL VII. ; or figures 63, 64, 65, 67, of PI. VIII., 
omitting the bolts. [The few non-isometric lines that occur, 
being in given isometric planes, can be located by a litle care.] 

2d. Construct the isometrical drawing of a low stout square- 
legged bench, and add the shadows. 

3d. Construct the isometrical drawing of a cattle yard with 
pens of various sizes and heights. 

4:th. Construct the isometrical drawing of a shallow partitioned 
drawer, or box, with the shadows. 

5th. Make the isometrical drawing of a cellar having a wall of 
irregular outline, and showing the chimneys, partitions, bins, etc. 



CHAPTER in. 

PEOBLBMS INVOLVING NON-ISOMETRICAL LINES. 

213. Since non-isometrical lines do not appear in their true size, 
each point in any of them, when it is in an isometric plane, must be 

ocated by two isometric lines, which, on the object itself, are at 
right angles to each other. Points, not in any known isometrical 
plane must be located by three such co-ordinates, as they are called, 
from a known point. 

214. Pkob. 5. To construct the isometrical draiving of the 
scarfed splice, shown at PL VIII., Fig. 66. Let the scale be y 1 -^ or 
three-fourths of an inch to a foot. In this case, PL XIII., Fig. 105, it 
will be necessary to reconstruct a portion of the elevation to the new 
scale (see PL XIII. , Fig. 106), where Ap=l% feet, pA" = 6 inches, 
and the proportions and arrangement of parts are like PL VIII., 
Fig. 66. In Fig. 105, draw AD, 3 feet; make DB and AA' each 
one foot, and draw through A' and B isometric lines parallel to AD. 
Join A — B, and from A lay off distances to 1, 2, 3, 4, equal to the 
corresponding distances on Fig. 106. Also lay off, on BK, the same 
distances from B, and at the points thus found on the edges of the 
timber, draw vertical isometric lines equal in length to those which 
locate the corners of the key in Fig. 106. Notice that opposite 
sides of the keys are parallel, and that AV, and its parallel at B, 
are both parallel to those sides of the keys which are in space per- 
pendicular to AB. To represent the obtuse end of the upper tim 
berof the splice, bisect AA', and make va=Aa, Fig. 106, and draw 
Aa and A'a. Locate m by va produced, as at ar, Fig. 106, and a 
short perpendicular=rm, Fig. 106, and draw mV and mV; VV 
being parallel to AA', and am being parallel to AV. To represent 
the washer, nut, and bolt, draw a centre line, vv\ and at t, the mid 
die point of Vn, draw the isometric lines tu and ue, which will giv* 
e, the centre of the bolt hole or of the bottom of the washer. A point 
— coinciding in the drawing with the upper front corner of the nut- 
is the centre of the top of the washer, which may be made } of an 
inch thick. 



96 PROBLEMS INVOLVING NON-ISOMETRICAL LINES. 

Through the above point draw isometric lines, rr' mApp', and 
lay off on them, from the same point, the radius of the washer, say 
2£ inches, giving four points, as o, through which, if an isometric 
square be drawn, the top circle of the washer can be sketched in it ? 
being tangent to the sides of this square at the points, as o, and 
elliptical (oval) in shape. The bottom circle of the washer is seen 
throughout a semi-circumference, i. e. till limited by vertical tan- 
gents to the upper curve. 

On the same centre lines, lay off from their intersection, the half 
6ide of the nut, 1 \ inches, and from the three corners which will be 
visible when the nut is drawn, lay off on vertical lines its thickness, 
1\ inches, giving the upper corners, of which c is one. So much 
being done, the nut is easily finished, and the little fragment of bolt 
projecting through it can be sketched in. 

Example.- — Construction of the nut when set obliquely. The nut 
is here constructed in the simplest position, i. e. with its sides in the 
direction of isometric lines. If it had been determined to construct 
it in any oblique position, it would have been necessary to have 
constructed a portion of the plan of the timber with a plan of the 
nut — then to have circumscribed the plan of the nut by a square, 
parallel to the sides of the timber — then to have located the cor- 
ners of the nut in the sides of the isometric drawing of the circum- 
scribed square. Let the student draw the other nut on a large scale 
and in some such irregular position. See Fig. 107, where the upper 
figure is the isometrical drawing of a square, as the top of a nut ; 
this nut having its sides oblique to the edges of the timber, which 
are supposed to be parallel to ca. 

215. Prob. 6. To make an isometrical drawing of an oblique 
timber framed into a horizontal one. PL XIII., Fig.108. Let the 
original be a model in which the horizontal piece is one inch square, 
and let the scale be J. 

Make ag and ah, each one inch, complete the isometric end of the 
horizontal piece, and draw ad, hn and gk. Let ab — "! inches, be — %\ 
inches, and draw bm. Let the slant of the oblique timber be such 
that if cd=2 inches, de shall be £ of an inch. Then by (213) de, 
parallel to ag, will give ce, which does not show its true size. Draw 
edges parallel to ce through b and m. In like manner/can be found, 
by distances taken from a side elevation, like PI. VIII. , Fig. 59. 

216. Prcxb. 7. To make an isometrical drawing of a pyramid 
standing upon a recessed pedestal. PL XIII., Fig. 109. (From a 



PROBLEMS INVOLVING NON-ISO METRICAL LINES. 97 

Model.) Let the scale be j. Assuming C, construct the isometric 
square, CABD, of which each side is 4£ inches. From each of its 
corners lay off on each adjacent side 1} inches, giving points, as e 
and/*/ from all of these points lay off, on isometric lines, distances 
of £ of an inch, giving points, as g, k, and h. From all these points 
now found in the upper surface, through which vertical lines can be 
seen, draw such lines, and make each of them one inch in length, 
and join their lower extremities by lines parallel to the edges of the 
top surface. 

Isometric lines through g, m, &c, give, by their intersections, the 
corners of the base of the pyramid, that being in accordance with 
the construction of this model, and the intersection of AD and CD 
is o, the centre of that base. The height of the pyramid — If inches 
— is laid off at ov. Join v with the corners of the base, and the 
construction will be complete. 

217. To find the shadows on this model. — According to prin- 
ciples enunciated in Division III., the shadow of fh begins at h, 
and will be limited by the line hs, s being the shadow off, and the 
intersection of the ray fs with hs. st is the shadow of/F. Accord- 
ing to (208), w is the shadow ofv, and joining w withp and q, the 
opposite corners of the base, gives the boundary of the shadow on 
the pedestal, ka is the boundary of the shadow of glc on the face 
ku. The heavy lines are as seen in the figure. If this drawing is 
to be shaded, the numerals will indicate the darkness of the tint for 
the several surfaces — 1 being the lightest. 

218. Prob. 8. To construct the isometrical drawing of a wall in 
batter, with counterforts, and the shadows on the wall. PI. XIII., 
Fig. 110. A wall in batter is a wall whose face is a little inclined 
to a vertical plane through its lower edge, or through any horizon- 
tal line in its face. Counterforts, or buttresses, are projecting parts 
attached to the wall in order to strengthen it. Let the scale be 
one fourth, i.e., let each one of the larger spaces on the scale 
marked 40, on the ivory scale, be taken as an inch. 

Assume C, and make CD = 2 inches and CI = 10 inches in the righ 
and left hand isometrical directions. Make DF=6 inches, EF=1^ 
inches, FG=10 inches, and GH=1|- inches, and join H and E, all 
of these being isometrical lines. Next draw EC, then as the top 
of the counterfort is one inch, vertically, below the top of the wall, 
while EC is not a vertical line, make F<^=one inch, draw de paral- 
lel to FE and ef, parallel to EH. Make ef and Ca each one inch, 
at /make the isometric Wnefg—^ of an inch, and at a make ab=2 



98 PROBLEMS INVOLVING NON-ISOMETBICAL LINES. 

inches, and draw bg. Make 5c=lJ inches, draw ch parallel to bg 
and complete the top of the counterfort. The other counterfort 
is similar in shape and similarly situated, L e., its furthermost lower 
corner, k, is one inch from I on the line IC, while HI is equal and 
parallel to CE. 

219. To find the shadows of the counterforts an the face of the 
\HiM, — We have seen—- PL IX., Fig. 78 — that when a line is per- 
pendicular to a vertical plane, its shadow on that plane is in the 
direction of the projection of the light upon the same plane, and 
from PL XII., Fig. 102, that the projection, an, of a ray on the left 
hand vertical face of a cube makes on the isometrical drawing an 
angle of 60° with the horizontal line ?m'. 

Now to find the shadow of mn. The front face of the wall — PL 
XIIL.Fig.llO— not being vertical, drop a perpendicular, fo, from an 
upper back corner of one of the counterforts, upon the edge ba y 
produced, of its base, and through the point o, thus found, draw a 
line parallel to CI. nfqp is then a vertical plane, pierced by mn, 
an edge of the further counterfort which casts a shadow; and 
np is the direction of the shadow of mn on this vertical plane. 
The shadow of mn on the plane of the lower base of the wall is of 
course parallel to mn, and^> is one point of this shadow, hence pq 
is the direction of this shadow. Xow n, where mn pierces the 
actual face of the wall, is one point of its shadow on that face, and 
q, where its shadow on the horizontal plane pierces the same face, 
is another point, hence nq is the general direction of the shadow 
of mn on the front of the wall, and the actual extent of this shadow 
is nr, r being where the ray mr pierces the front of the wall. 

From r, the real shadow is cast by the edge, mu, of the counter- 
fort, r, the shadow of m, is one point of this shadow, and 5, where 
um produced, meets yn produced, is another (in the shadow pro- 
duced), since s is, by this construction, the point where um, the 
line casting the shadow, pierces the surface receiving the shadow. 
Hence draw srt, and nrt is the complete boundary of the shadow 
sought. The shadow of the hither counterfort is similar, so far as 
it falls on the face of the wall. 

Other methods of constructing this shadow may be devised by 
the student. Let t be found by means of an auxiliary shadow of vm 
on the plane of the base of the walL 

Remark. — In case an object has but few isometrical lines, it is 
most convenient to inscribe it in a right prism, so that as many of 
its edges, as possible, shall lie in the faces of the prism. 



CHAPTER IV. 

PROBLEMS INVOLVING THE CONSTRUCTION AND EQUAL DIVISION Of 
CIRCLES IN ISOMETRICAL DRAWING. 

220. Prob. 9. To make an exact construction of the isometncal 
drawing of a circle. PL XIII. , Pigs. 111-112. This construction is 
only a special application of the general problem requiring the con- 
struction of points in the isometric planes. 

Let PI. XIII., Fig. Ill, be a square by which a circle is circum- 
scribed. The rhombus — Fig. 112 — is the isometrical drawing of the 
same square, CA being equal to C'A'. The diameters g'h' and e'f 
are those which are shown in their real size at gh and ef giving 
<7, h, e, and fas four points of the isometrical drawing of the circle. 
In Fig. Ill, draw b'a ', from the intersection, b\ of the circle with 
AT)' and parallel to C'A'. As a line equal to b'a', and a distance 
equal to A' a' can be found at each corner of Fig. Ill, lay off each 
way from each corner of Fig. 112, a distance, as Aa, equal to A'«', 
and draw a line ab parallel to CA and note the point b, where it 
meets AD. Similarly the points n, o, and r may be found. Having 
now eight points of the ellipse which will be the isometrical draw- 
ing of the circle, and knowing as further guides, that the curve is 
tangent to the circumscribing rhombus at g, h, e and /, and perpen. 
dicular to its axes at b, n, o and r, this ellipse can be sketched in by 
hand, or by an irregular curve. 

221. If, on account of the size of the figure, more points are desira- 
ble they can readily be found. Thus ; on any side of Fig. Ill, take 
i distance as CV and c'd' perpendicular to it, and meeting the circle 
at d'. In Fig. 112, make Cc=CV, and make cd equal to c'd' and 
parallel to CD, then will d be the isometrical position of the 
point d'. 

222. Prob. 10. To make an approximate construction of the iso 
metrical drawing of a circle. PI. XIII. , Fig. 113. By trial we shall find 
that an arc, gf having C for a centre and Of for its radius, will very 
nearly pass through n ; likewise that an arc eA, with B for a centre, 
will very nearly pass through r. These arcs will be tangents to the 



100 PROBLEMS INVOLVING THE ISOMETRIC AL DRAWING OF CIRCLES. 

sides of the circumscribing square at their middle points, as they 
should be, since C/and Be are perpendicular to these sides at their 
middle points. Now in order that the small arcs, fbh and goe, should 
be both tangent to the former arcs and to the lines of the square at 
g, A, f and e, their centres must be in the radii' of the larger arcs, 
hence at their intersections p and q. Ares having jt? and q for cen- 
tres, and ph and qe as radii, will complete a four-centred curve 
which will be a sufficiently near approximation to the isometrical 
ellipse, when the figure is not very large, or when the object for 
which it is drawn does not require it to be very exact. 

223. Prob. 11. To make an isometrical drawing of a solid com- 
posed of a short cyli?ider capped by a hemisphere. PL XIII., Fig. 114. 
Scale=i. Let this body be placed with its circular base lowermost, 
as shown in the figure. Make ac and bd the height of the cylin- 
drical part=l T 5 T inches, and draw cd. Now a sphere, however 
looked at, must appear as a sphere, hence take e, the middle point 
of cd, as a centre, and ec as a radius, and describe the semicircle 
ehd, which will complete the figure. 

224. In respect to execution, in general, of the problems of this 
Division, a description of it is not formally distinguished from that 
of their construction, since the figures generally explain themselves 
in this respect. In the present instance, the visible portion of the 
only heavy line required will be the arc anb. As there is no angle 
at the union of the hemisphere with the cylinder — see the preceding 
problem — no full line should be shown there, but a dotted curve 
parallel to the base and passing through c and d, might be added to 
show the precise limits of the cylindrical part. 

225. Again, if it be desired to shade this body, the element, ny, 
of the cylindrical part, with the curve of shade, pfry, on the sphe- 
rical part, will constitute the darkest line of the shading. The curve 
of shade, pfry, is found approximately as follows. The line ny being 
the foremost element of the cylinder, yy'=ne, is the projection of 
an actual diameter of the hemisphere, mm" and gg", parallel to ST, 
are the radii of small semicircles of the hemisphere, to which projec- 
tions of rays of light may be drawn tangent, and m' and g\ are 
the true positions of their centres — ym' being equal to nm, and 
yg' equal to ng. Drawing arcs of such semicircles, and drawing 
rays, fd and rS, tangent to them, we determine f and r, points of 
the curve of shade on the spherical part of the body, through 
which, with p and y, the curve may be sketched. 

226. Prob. 12. To construct the isometrical circles on the thres 



p:l.xiii 




PROBLEMS INVOLVING THE ISOMETRICAL DRAWING OF CIRCLES. 101 

visible faces of a cube, as seen in an isometrical drawing. Pi, 
XIV., Fig. 115. This figure needs no minute description here, 
being given to enable the student to become familiar with the 
position of isometrical circles in the three isometrical planes, and 
with the positions of the centres used in the approximate construc- 
tion of those circles. By inspection of the figure, the following 
general principle may be deduced. The centres of the larger arcs 
are always in the obtuse angles of the rhombuses which represent 
the sides of the cube, and the centres of the smaller arcs are at the 
intersection of the radii of the larger arcs with the diagonals 
joining the acute angles of the same rhombuses — i. e. the longer 
diagonals. 

227. Prob. 13. To make the isometrical drawing of a bird house. 
PL XIV., Fig. 116. Assuming C, make CA' = 16 inches, Ca=3 inches, 
and CB = 9 inches. At «, make ab—1 inch, and ac — 8 inches. Draw 
next the isometric lines BD and cD. Through b make 5E = 1 6 inches, 
make EA=3 inches, and EF=7 inches. Then draw the isometric 
lines DH and FH. Bisect b~E at N, make N/=ll^ inches, draw 
cf and F/j make ce=¥h=fg=OTie inch, and draw eg and hg. 
Through A and b draw isometric lines which will meet, as at a'. 
On £E make bJe, Im and nE each equal to 3 inches, and let Jel and 
mn each be 2>\ inches. At I and n draw lines, as Iv, parallel to CB, 
and one inch long, and at their inner extremities erect perpendicu- 
lars, each 3^- inches long. Also at Jc, I, m and n, draw vertical iso- 
metrical lines, as Jet, 3|- inches long. The rectangular openings thus 
formed are to be completed with semicircles whose real radius is 
1£ inches, hence produce the lines, as Jet — on both windows — mak- 
ing lines, as JcG, 5| inches long, and join their upper extremities as 
at GI. The horizontal lines, as ts, give a centre, as s, for a larger 
arc, as tu. The intersection of Go with Iz — see the same letters on 
Fig. 115 — gives the centre, p, of the small arc, uo. The same ope- 
rations on both openings make their front edges complete. Make 
oq and pr parallel, and each, one inch long, and r will be the centra 
of a small arc from q which forms the visible part of the inner edge 
of the window. Suppose the corners of the platform to be rounded 
by quadrants whose real radius is 14- inches. The lines a'b and 
bJc each being 3 inches, h is the centre for the arc which repre- 
sents the isometric drawing of this quadrant, whose real centre 
on the object, is indicated on the drawing at y. So, near A, w ia 
the centre used in drawing an arc, which represents a quadrant 
whose centre is x. — See the same letters on Fi<r. 115. 



102 PROBLEMS INVOLVING THE ISOMETRICAL DRAWING OF CHICLES. 

Of the Isometrical Drawing of Circles which are divided in 
Equal Parts. 

228. Peob. 14. PI. XIY., Fig. 117. First method.— If the semi 
ellipse, ADB, be revolved up into a vertical position about AB as an 
axis, it will appear as a semicircle AD'B of which ADB is the iso- 
metrical projection. Since AB, the axis, is parallel to the vertical 
plane, the arc in which any point, as D, revolves, is in a plane per- 
pendicular to the vertical plane, and is therefore projected in a 
straight line DD'. Hence to divide the semi-ellipse ADB into parts 
corresponding to the parts of the circle which it represents, divide 
AD'B into the required number of equal parts, and through the 
points thus found, draw lines parallel to D'D, and they will divide 
ADB in the manner required. The opposite half of the curve can 
of course be divided in a similar manner. 

229. Second method. — CE is the true diameter of the circle of 
which ADB is the isometrical drawing. Let it also represent the 
side of the square in which the original circle to be drawn is inscribed. 
The centre of this circle is in the centre of the square, hence at O, 
found by making eO equal to half of CE, and perpendicular to that 
line at its middle point e. 

With O as a centre, draw a quarter circle, limited by CO and EO, 
and divide it into the required number of parts. Through the points 
of division, draw radii and produce them till they meet CE. CE, 
considered as the side of the isometrical drawing of the square, is 
the drawing of the original side CE of the square itself with all 
its points 1, 2, 6, 7, &c, and O' is the isometrical posi- 
tion of 0. Hence connect the points on CE with the point O r and 
the lines thus made will divide the quadrant BC in the manner 
required. 

Applications of the preceding Problem. 

230. Peob. 15. To make an isometric drawing of a segment oj 
an Ionic Column. PL XIY., Fig. 118. Let dD be a side of the cir- 
cumscribing prism of the column. By the second method of Prob. 
14, find O', the centre of a section of the column, and with O' as a 
centre, draw any arc, as a'q' . The curved recesses in the surface 
of a column are called flutes, or the column is said to be fluted. In 
an Ionic, and in some other styles of columns, the flutings are semi- 
circular with narrow flat, or strictly, cylindrical surfaces, as ee"p, 
between them. Hence, in Fig. 118, assume a'b\ equal to q'v\ as 
half of a space between two flutes, divide b'v' into four equal parts, 
*nd make the points of division central points of the spaces as f'e f 



PROBLEMS INVOLVING THE ISOMETRICAL DRAWING OF CIRCLES. 103 

between the flutes. Let the flutes be drawn with points, as c' a? 
centres and touching the points as b'd' ; then draw an arc tangent, 
as at r, to the flutes. To proceed now with the isometrical draw- 
ing, draw, in the usual way, the isometrical drawing of the outer 
circumferences of the column, tangent to aD and b'"¥ — assuming" 
DF for the thickness of the segment. Now a'q' being any arc, and 
not one tangent to aD so as to represent the true size of a quadrant 
of the outer circumference, the true radius of the circle tangent to 
the inner points of all the flutes will be a fourth proportional, O'y' 
to O'f, Oi (=0'y), and O's. On Oi\ lay off OY=Oy', draw IJ to 
find a centre I, and similarly find the other centres of the larger arcs 
of the inner ellipse. The points n, h and n\ h! are the centres of the 
small arcs (222) for the two bases. Having gone thus far, produce 
0'b\ O'c', &c. to dD ; at b, c, &c, erect vertical lines, W", cc"\ 
<fcc., then from 5, c, &c. draw lines to O, and note their intersec- 
tions, b", c" ', &g. with the curves of the lower base ; and from 
b"\ c'", &c. draw lines to O" and note their intersections, b"", c'"'\ 
&c. with the ellipses of the upper base. This process gives three 
points for each flute by which they can be accurately sketched in, 
remembering that they are tangent to the inner dotted ellipses, as 
at c"", 0'", &c. and to the radii, as e"0" — at e" . Parts beyond 
FO" are projected over from the parts this side, thus drawn. 

231. Prob. 16. To construct the isometrical drawing of a seg- 
ment of a Doric Column. PI. XIV., Fig. 119. The flutes of a Doric 
column are shallow and have no fiat space between them. Adopt- 
ing the first method of Prob. 14, let the centre, A, of the plan be 
in the vertical axis, GA', of the elevation, produced. Let Ac and 
Kb be the outer and inner radii containing points of the flutes. 
Make Ad=% of Ac, for the radius of the circle which shall contain 
the centres of the flute arcs. Let there be four flutes in the qua- 
drant, shown in the plan. Their centres will be at A, <fcc, where 
radii A<7, <fcc, bisecting the flutes, meet the outermost arc. In pro 
ceeding to construct the isometrical drawing, project b and c, at b 
and c' on the axis A'd'. Now, owing to the variation at b and c' 
between the true and the approximate ellipse, we cannot make use 
of the latter, if we retain b' and c' in their proper places, as projected 
from b and c, hence through V and c' draw isometric lines which 
locate the points W and Q' (the points are between these letters) 
which are the true positions of N and Q respectively. Correspond- 
ing points, between N'" and y, are similarly found. By an irregular 
curve the semi-ellipses vb'Q' and N"VN' can be quite accurately 



104 PROBLEMS INVOLVING THE ISOMETRICAL DKAWING OP CIRCLES. 

drawn. Next, project upon these curves the points w, e, &c, r, <7, 
&c. of the flutes — as at u', e\ &c, r', &c, and with an irregular 
curve draw the curves through these points, tangent to the inner 
semi-ellipse. The corresponding curves of the lower base are found 
by drawing lines r'r'\ u'u", &c. through the points of tangency, 
r', h\ &c, and through u\ &c, and all equal to FD, the thickness 
of the segment. The curves above the axis K'd! are projected 
across from those already made below it. Let this figure and the 
last be made separately on a very large scale. 

Special Examples. 

232. Prob. IV. To draw a cube or other parallelopipedical body 
60 as to show its under side. PL XIV., Pig. 120. By reflection, it 
becomes evident that it is the relative direction of the lines of 
the drawing among themselves, that make it an isometrical draw- 
ing. Hence in the figure, where all the lines are isometric lines, the 
whole is an isometric drawing, now that the solid angle C is nearest 
us, as much as if the angle A (lettered C on previous figures) were 
nearest us. 

233. Remark. By a curious exercise of the will, we can make 
Fig. 120 appear as an interior view, showing a floor CFED, and 
two w T alls; or, in Fig. 115 and others, we can picture to ourselves 
an interior showing a ceiling GI& and two walls. This is probably 
because — 1st. All drawings being of themselves only plane figures, 
we educate the eye to see in them, what the mind chooses to conceive 
of, as having three dimensions. 2nd. When, as in isometrical draw- 
ing, the drawing in itself as a plane figure, is the same for an interior 
as for an exterior view of any given magnitude, the eye sees in it 
whichever of these two the mind chooses to imagine. 

234. Prob. 18. To construct isometrical drawings of oblique 
sections of a right cylinder with a circular base. PL XIV., Fig. 
121. This construction is easily made from a given circle as a base 
of the cylinder, that base being in an isometric plane. The circle in 
the plane AGEF is such a circle. Let A'G'E'F' be a plane inclined 
to AGEF but perpendicular, as the latter is, to the planes GB and 
DF, and let A'G'E'F" be a plane inclined to all the sides of the 
prism AGE— D. 

Lines, as aa'a\ &c, being in the faces of the prism and parallel 
to their edges, meet the intersections, F'E'— F"E", &c. of the 
oblique planes at points a, a", <fcc, which are points of oblique 
Bections of a cylinder inscribed in the prism AGE — D, and wnose 
base is acbdu. 



PROBLEMS INVOLVING THE ISOMETK1CAL DRAWING OF CIRCLES. 105 

So, points, as c, have the corresponding points cV, &c. in the 
diagonals A'E', A*E' of the planes in which those points are found. 

To find points, as t', t\ &c. corresponding to t in the base, draw 
any line, as yd, through £, and find the corresponding lines, as y'oV 
and y"d". Their intersections with the diagonals G'F' and G"F* 
will give the points t', t\ &c. Having thus found eight points of 
each oblique section of the given inscribed cylinder whose base is 
abcd-u, and remembering that each of these sections is tangent to the 
sides of its circumscribing polygon (considering the lines y'd\&c.) y 
the curves a\ b,' c', t\ and a", b\ c", t" are readily sketched in. • 

235. Remarks, a. As before stated, it is the relative direction, 
among themselves, of the lines of an isometrical drawing, that deter- 
mine it as an isometrical drawing, hence PL XIV.,, Fig. 121/isan 
isometrical drawing, though its lines are not situated with refer- 
ence to the edges of the plate as the similar lines of previous 
figures have been. If the portion of the plate containing this figure 
were cut out so as to make the edges of the fragment, so cut out, 
parallel and perpendicular to GE, the figure would appear like 
the previous isometrical drawings. 

b. The problem just solved must not be confounded with one 
which should seek to find the isometric projection of a curve 
which in space is a circle on the plane G'E' — A', for the curve 
a'b'c'd't' is not a circle, in space. 

236. Peob. 19. To solve the problem just enunciated. PL XIV., 
Figs. 121-122. e"r — Fig. 122— is a plan of the section rA'F' in 
which — it being a square — a circle can be inscribed. e"r is there- 
fore the plan of the circle also. Making rG — Fig. 122 — equal to 
rG' — Fig. 121, and drawing e"G, we have the plan of the section 
G'E' — A', and making o"p', Fig. 122, equal to eV, we have the plan 
of a circle in the section G'E' — A'. Now draw o'x and £>V — Fig. 
122 — make A'e and GV" and e'"p and eo — Fig. 121 — equal to e"x, 
Ge\ e'p' and xo"— Fig. 122— draw pY and oU; and u'JJ and b'Y 
to intersect them, and we shall have U and Y as the isometric 
positions in the plane G'E' — A' of the points o' and p' which, 
considered as points on the circle, are evidently enough extre- 
mities of its horizontal diameter, at which points, the circle is tan- 
gent to the vertical lines whose isometric positions in the plane 
G'E' — A' are pY and oU. T and a' are other points. 

The finding of intermediate points, which is not difficult, is left a* 
an exercise for the student. 



CHAPTER V. 



OBLIQUE PROJECTIONS. 



x. There is a kind of projection, examples of which, in the draw 
in.g of details, etc., are oftener seen in French works than isometri 
cal projection (an English invention) is. It has been variously 
named, "Military," "Cavalier," or "Mechanical" Perspective. It 
may be called " Cabinet Projection," it being especially applicable 
to objects no larger than those of cabinet work, and being actually 
used in representing such work. It is properly called oblique 
projection, because in it the projecting lines, which have been 
hitherto made perpendicular to the plane of projection are oblique 
to that plane ; and pictorial projection, on account of its pictorial 
effect, as seen in PI. I., Figs. 1, 2, 3, etc. 

2. This new projection differs from isometrical, chiefly in show- 
ing two of the three dimensions of a cube, for example, in their true 
direction as well as size. 

Thus; Fig. 1 is the isometrical drawing of a cube, and Fig. 2 is 




4/ 

Hi 

/ 

£ — ( 



Fig. 2. 



an oblique projection of the same cube ; all the edges being of the 
same length in both figures. Hence we see, as stated, that in the 
latter figure, the faces DEFG, and ABCH, and by consequence 
every line in them, are shown in their true form, as well as size ; 
which is not true of isometrical drawing. 



OBLIQUE PROJECTIONS. 107 

3. Another advantage of oblique projection, already apparent, is, 
that the remote corner, H, which, in the isometrical drawing of a 
cube, coincides with the foremost corner, G, is seen separately in 
the oblique projection. 

Also, of the four body diagonals of the cube, one, GH, appears 
as a point only, in isometrical projection, and the other three, as 
FC, are all partly confounded with the projections, as FG, of edges. 
But in the oblique projection, all these diagonals show as lines, and, 
except BE, separately from the edges of the cube. 

4. In the projections hitherto considered, the projecting lines of 
a point have uniformly been taken perpendicular to the planes of 
projection. 

Sometimes, however, the projecting lines, or direction of vision 
(12) are oblique to the plane of projection. 

There are thus two systems of projection in which the eye is at 
an infinite distance. First, common or perpendicular projections, 
in which the projecting lines (5) are perpendicular to the plane of 
projection. Second. Oblique projections, in which those lines are 
oblique to the plane of projection. 

Isometrical, is a species of perpendicular projection. We shall 
now proceed to explain the simple and useful form of projection 
which is called oblique projection. 

5. If a line, AB, Figs. 3, or 3a, be perpendicular to any plane 
PQ, its projection on that plane, in common projection, would be 
simply the point B. But if we suppose the projecting line, AC, of 
any point, A, to make an angle of 45° with the plane of projection 
PQ, it is evident that the projection of A would be at C, and the 
projection of AB on PQ would be BC; also that BC=AB. That 
is, the projection, as BC, of a perpendicular to the plane of projec- 
tion is equal to that perpendicular itself 

Any line through A and parallel to the plane PQ would evidently 
be projected in its real size on PQ. Hence, finally, the system of 
oblique projections here described, allows us to show the three 
dimensions of a solid in their real size, on a single figure ; but only 
ptarallels, and perpendiculars to the plane of projection, appear in 
their true size. 

6. It is now evident from Fig. da, that there may be an infinite 
number of lines from A, each making an angle of 45° with the 
plane PQ. These lines, taken together, would form a right cone 
with a circular base, whose axis would be AB, whose vertex would 
be A, and whose base would be a circle in the plane PQ, drawn 
with B as a centre, and BC as a radius. Each radius of this circle 



108 



OBLIQUE PROJECTIONS. 



would be an oblique projection of AB, corresponding with the 
element as C'A, from its extremity, taken as the direction of the 
projecting lines. That is, the oblique projection of AB may he 
drawn equal to AB and in any direction. 

1. Thus Figs. 4, 5, 6 and many more are all equally oblique projec- 
tions of the same cube. The paper represents the plane of projec- 
tion ; FA is perpendicular to the paper at A. The eye, relative to 
Fig. 2, is looking from an infinite distance above and to the right 
of the body, and in a direction making an angle of 45° with the 
paper. And, generally, in oblique projection, the direction of 
vision = the projecting lines, may have any direction (the same for 
all points in the same problem) making an angle of ±5° with the 
plane of projection. Hence FA may have any direction relative 
to FG and yet be always equal to FG, that is to the original of FA 
in space. Thus, CDK = 45°, in Fig. 4 ; 30°, in Fig. 5; and 60°, in 
Fig. 6. Also, DC, FA, etc., may incline to the left, or downward. 




Fig. 3. 



Pig. 3a. 



/ 


/ 




c 


hL..£ 


~ 7 


/ $ 


/ 


'' » 1 ' V ' I ' . 


/ 



E ^a D 

Fig. 4. 







y^ 


H, 




c 









D 
Fig. 5. 



OBLIQUE PROJECTIONS. 



109 



In Fig. 6, CDK = 60°. Accordingly DK=|DC, from which, 
having found K, the perpendicular KC can be drawn to limit DC. 
Or, as before, CK=i v% DC being =1. That is, CK = |EC, since 
CDE = 120°. And for a square prism of any length, KC= half of 
the diagonal joining alternate vertices of a regular hexagon whose 
side equals the edge of the prism, lying in the direction of DC, and 
whose length is supposed to be given. 



F 


A 






£ 


HI 

/ 


E 


[ 


) 



Fig. 6. 

With these illustrations, the student might proceed to investi- 
gate other relations between the parts of these, and still other 
oblique projections. But the above may suffice for now. 

We observe that, in Figs 5 and 6, none of the body diagonals 
are confounded with the edges ; and that each of the tnree forms 
may be preferable for certain objects. 

8. Points not on the axes EF, EH, and ED, or on parallels to 
them, are found by co-ordinates, as in isometrical drawing. Thus, 
if ED, Fig. 4, be 4 inches, and if we make Ea=2 inches, ab paral- 
lel to EH, = 2 J inches, and be, parallel to EF, = 1J inches, then c is 
the oblique projection of a point, 2 inches from the face FH ; 2\ 
inches from the face FEG, and 1 \ inches above the base EDC. 
This principle will enable the student to reconstruct any of the 
preceding isometrical examples of straight-edged objects, in ob- 
lique projection. 

9. It only now remains to explain the obliqiie projections of cir- 
cles. Let Fig. 7 be the oblique projection of a cube, with circles 
inscribed in its three visible faces. One of these circles, abed, will 
appear as a circle, and so would the invisible one on the parallel 
rear face. 

For the ellipse in BCDG, draw the diagonals, BD and CG, of 



110 



OBLIQUE PROJECTIONS. 



that face. Then in the cube itself, horizontal lines joining corre- 
sponding points in the circles abed, and hpfu, are parallel to the 
diagonal EC. Hence tu, ef, mp, and gh determine the points u,f,p 




Kg.*. 

and A, by their intersections with the diagonals BD and CG. The 

middle points of the sides of the face BCDG, are also points of the 
ellipse, and are its points of contact with those sides. The ellipse 
also has tangents at h and/*, parallel to BD, and at u and />, paral- 
lel to CG. Hence, having eight points, all of which are points of 
contact of known tangents, the ellipse can be accurately sketched. 

10. The ellipse in the upper face could be found in the same 
manner. But an approximate construction by circular arcs has 
been shown, to test its accuracy and appearance, as compared with 
the approximate isometrical ellipse. The ellipse being tangent at 
L and K, perpendiculars to FA and BA, at those points, will in- 
tersect at M, the centre of an arc tangent to FA and BA at those 
points. Then «X perpendicular to FG at a, and equal to MK, 
gives N, the centre of the arc an. As the remaining arcs must be 
tangent to those just drawn, their centres, r and *, must be the 
intersections of the radii of the large arcs, with the transverse axis, 
FB, of the ellipse. 

The true extremities of the transverse axis are found by drawing 



OBLIQUE PROJECTIONS. Ill 

pq parallel to AC, and a parallel to it from u. The error qq' at 
each end of the transverse axis, is thus seen to be considerable 
Also the greater difference between the radii, than occurs in mak- 
ing the isometrical ellipse, occasions a harsh change of curvature at 
K, N, etc. ; so that the approximate construction of the oblique 
ellipse is of very little value. 

11. It is found on trial, that the centre M falls both on BD and 
EC, so that neither KM nor LM really need be drawn. The reason 
of this property, which so simplifies the construction, is evident. 
For, BA=AF^FE are in position as three sides of a regular octa- 
gon, so that the perpendiculars, as KM, from the middle points of 
those sides, will meet at the same points with BD, AGM, EC, etc., 
which are obviously the bisecting lines of the angles of the octa- 
gon, viz. at the centre, M, of the octagon. 

By varying the angle GFA, as in the previous figures, the stu- 
dent may discover similar coincidences, which he can explain for 
himself. 

Finally, it is to be noticed, that the pictorial diagrams of PI. L, 
Figs. 1, 2, 3, 5, etc., which are so effective a substitute for actual 
models, to most eyes, are merely oblique projections of models 
themselves. 

Practical Examples. 

12. PI. XV. shows some further illustrations of oblique projec- 
tion in contrast with isometrical drawing. 

Fig. 1 is an isometrical drawing of a roller and axle, showing the 
parallel circumscribing squares of its several parallel circles; and 
the circumscribing prism, mnpo, of the roller, placed so as to show 
its lower base. The distances ab, bd and de, between the centres 
of the circles, are thus seen in their true size, in this, and on the 
next two figures. Also the several circles and their centres, have 
the same letters on the same figures. 

Fig. 2 shows an oblique projection of the same object, when its 
axis, «e, is made perpendicular to the paper. This is the simplest 
position to give to the object, since its several circles, being thee 
parallel to the paper, will appear respectively as equal circles in 
the figure. And, generally, in making oblique projections of ob- 
jects having some circular outlines, the object should be so placed, 
that the majority of these outlines should be in planes parallel to 
the plane of projection. 

Example. — Make ae in any other direction. 

Fig. 3 shows another oblique projection of the same object, but 



112 OBLIQUE PROJECTIONS. 

with its axis ae parallel to the paper, or plane of projection. Dif- 
ferent wheels and their axles in the same machine, might have the 
two positions indicated in Figs. 2 and 3. Hence it is necessary to 
understand both; though if drawing only a single object of this 
kind, we should for convenience make it as in Fig. 2, only remem- 
bering, as explained in previous principles, that ae may be drawn 
in any direction. 

Examples. — 1st. In Fig. 1, let the upper end of the axis be the 
visible one. 

2d. In Fig. 3, let ae be horizontal and parallel to the paper and 
let the left hand end of the body be seen. 

PL XV., Figs. 4, 5, and 6 are a plan and two isometrical draw- 
ings, in full size, of a hexagonal nut. Fig. 4 is the plan of the nut 
with the circumscribing rectangle, mnop, containing two of its sides, 
CD and AF. Fig. 5 is the isometrical drawing of the same, and 
thus shows the face CD A, in its true size. The edges B£, Cc, etc., 
and centre heights, HA, of the faces, also show in their real size, as 
does the height Oo of the nut. BC is greater than its real size, 
being more nearly parallel to pn than pm is. AB is less than its 
true size, AB, Fig. 4; being nearer perpendicular to pn than pm is. 

Fig. 6 is, perhaps, a more agreeable looking isometrical figure 
of the nut, but it shows only the heights in their true sizes, except 
as pq equals the diameter of the circumscribing circle MNL of the 
hexagonal base of the nut, so that half of pq equals the true width 
of the faces. 

This figure makes an application of (Prob. XIV., First Method). 
Thus, having made the isometrical circle MNKL, in the usual way, 
describe the semicircle MN'K'L on ML as a diameter, and inscribe 
the semi-hexagon in it with vertices as M, N', K' and L. Then 
by revolving the semicircle back to its isometrical position, N' and 
K' will fall at N and K. 

The surface Q, in Figs. 5 and 6, represents a spherically rounded 
surface of the nut, while the surface, R, is plane. By finding three 
points as c, h and D, Fig. 5, in each upper edge of a face, those 
edges can be drawn as circular arcs; and the visible boundaries, 
qq, of the rounded surface Q can be sketched, as indicated. 

Examples. — 1st. Make oblique projections corresponding to Figs. 
5 and 6. 

2d. Also with the top, R, of the nut parallel to the plane of pro- 
jection, and either in isometrical or oblique projection. 

dd. Also as if Fig. 6 were turned 90° about Oo, so as to show 
only two faces of the nut. 



OBLIQUE PROJECTIONS. 113 

Figs. 7, 8, and 9 show a plan and oblique projection of a model 
of an oblique joint. 

Fig. 7 shows, once for all, that in every case of oblique^ as well 
as of isometrical drawing, where the lines as dg and gp, of the ob- 
ject, are oblique to each other, the body must be conceived to be 
inclosed in a circumscribing rectangular prism, whose sides shall 
contain its points, or from which they can be laid off by ordinates 
as mo, parallel or perpendicular to those sides. 

Fig. 7 is on a scale of one half, and Fig. 9 is in full size. Then, 
supposing the scale to be the same Cn, Fig. 9, = en Fig. 7, mo, 
wP, e'A, etc., in Fig. 9 = mo, np, ih, etc., in Fig. 7. So fe and/s, 
Fig. 9 = the same in Fig. 7. 

Thus the edges of timber A are shown in their real size, but 
those of B are distorted by their position. B is separately shown 
in its true proportions in Fig. 8, that is so far as its lines are paral- 
lel to ko, oO or op. 

Of the heavy lines in Oblique Projection. 

These simply follow the same rule, relative to the given object 
that is applied in common, or perpendicular projections; (16-20). 
• Thus, in PL XV., Fig. 2, the semicircles of A, B and C below 
ae would be heavy, and the opposite parts of D, and E. Also if B, 
Fig. 8, represents a timber parallel to the ground line, the heavy 
lines would be as there shown. And likewise on Fig. 9, where 
these lines are indicated by double dashes across them. 

In short, conceive the common projections of an object to be 
given with the heavy lines drawn. The oblique projection of the 
same object, placed in the same position, would simply show the 
oblique projections of the same heavy lines. That is, the same 
iines would be heavy in both kinds of projection. 



DIVISION FIFTH 

ELEMENTS OF MACHINES. 



CHAPTEE I. 

PRINCIPLES. SUPPORTERS AND CRANK MOTIONS. 

General Ideas. 

1. Machines generally effect only physical changes. That is,, 
they are designed to change either the form or the position of 
matter. They do this either directly, as in machines that ope- 
rate immediately on the raw material to be wrought, as looms, 
lathes, planers of wood or metal, etc., or indirectly, as in the 
machines called prime movers, like steam-engines and water- 
wheels, which actuate operating machines. 

We have, then, Prime movers and Operative machines. Also, 
of the latter, machines for changing the position of matter, as 
pumps, cranes, etc. ; and machines for changing its form, as 
lathes, planers, etc. ; and each with many subdivisions. 

2. In every machine there are to be distinguished the sup- 
porting parts, which are generally fixed and rigid, and the 
working parts, which are moving pieces. 

The supporting parts are general, supporting the entire ma- 
chine; or local, supporting some one part, as the px 'How -block, also 
called a plummer-block, or a pedestal, which supports a revolv- 
ing shaft ; or the guide oars, plainly seen in some form at the 
piston-rod end of the cylinder of any locomotive or other steam- 
engine, and which, by means of the stout block, called a cross- 
head, sliding between them, constrain the piston-rod, which is 
fastened to the cross-head, to move in a straight line. 

3. The working parts are connected together, forming a train, 
subject to this law, that a given position of any one piece deter- 
mines that of all the others. For the purpose of making the 
drawing of a machine, it is not enough, therefore, only to take 



PL. XIV. 




SUPPORTERS AND CRANK MOTIONS. 11& 

the measurements of its parts. This will suffice for the frame, 
but the motions of the train must be understood, so as to know 
what position to give to other parts, corresponding to a given 
position of some one part. 

4. In some machines, however, there are subordinate trains, 
serving to adjust the position or speed of the principal trains, as 
in case of engine governors. Also some parts are adjustable 
by hand, as the position of the bed in a drilling machine, or of 
the tool and rest containing it, in a lathe. 

5. The working parts of every machine consist of certain me- 
chanical elements or organs, which are comparatively few in 
number, and not always all present in any one machine. The 
principal of these are pistons, cross-heads, shafts, cranks, cams 
and eccentrics, toothed wheels, screws, band-pulleys, connecting- 
rods, bands or chains, sliding or lifting valves, grooved links, 
rocking arms and beams, flat or spiral springs, chambered parts 
and internal passages, as pump-barrels, steam-cylinders, valve- 
chests, etc. 

6. These, considered separately, are of various degrees of com- 
plexity of design, many of them quite simple. By far the most 
important, relative to the geometrical theory of their perfect 
action, are toothed wheels of various forms. These we shall 
therefore principally consider, together with a few other useful 
examples. 

Supporters. 

Example 1. A Pillow-block. Pillow-blocks of various 
designs, adapted to horizontal, or vertical, or beam engines, 
are so common, and so generally represented in works on prac- 
tical mechanism, that the following figure, taken from a drawing 
to scale, is inserted here as a sufficient guide; the object, more- 
over, being symmetrical with respect to the centre line 00', and 
a little more than half shown. 

Description. — BB', not definitely shown in plan, is a portion 
of the main bed of an engine. SS' is the sole of the pillow- 
block, DD' its body, CC its cover, and del — d r d' the brasses 
which immediately enclose the fly-wheel shaft of the engine. 
The holding-down bolts, as h — b'~b\ pass through slotted holes 
pq, a little wider, that is, in the direction pq than the diameter 



SUPPORTERS AA T D CRANK MOTIONS. 117 

of the bolt. This construction allows for adjustment of the 
position of the block by wedges, driven between the sole and 
stops I, solid with the bed B'. 

The cover CO' is held in place by bolts c — c'W , the heads h r 
being in recesses, sunk in the under side of the sole. The spaces 
at r and n between the cover and the body, allow for the wear 
of the brasses del — cl'cl' against the shaft. To prevent lateral or 
rotary displacement of the brasses, ears ee — e'e' project from 
them into recesses in the cover and body of the block. The same 
end is often attained by making their outer or convex surfaces 
octagonal, and by providing them with flanges where they enter 
and leave the block. 00' is the oil cup, here solid with the 
cover, but oftener now a separate covered brass cup, contrived 
to supply oil gradually to the shaft. 

Construction. — The 'proportions of the figure being correct, 
assume eig, the half length of the sole, to be 12 inches, and meas- 
ure by a scale its actual length on the figure. A comparison of 
the two will indicate the corresponding scale of the figure.* 

Then, having determined the scale, all the other measure- 
ments can be determined by it to agree with each other, and the 
figure can be drawn on any scale desired, from J to -J- of the full 
size. 

The body being symmetrical, all the measurements to the left 
from 00' can be laid off to the right of it, and the complete 
projections thus constructed. 

The method of drawing hexagonal nuts has been shown in 
detail in Div. L, Problems 31, 32. 

Execution. — Note the heavy or shade lines as in previous ex- 
amples (Div. II.); but if the figure is to be shaded and tinted, ink 
it wholly in pale lines or none. 

Exercises. — 1. Construct from the two given projections an end eleva- 
tion of the block. 

2. Construct a vertical section on the centre line Ob. 

3. Construct a top view with the cover removed. (The dotted lines 
showing the internal construction will enable these sectional views to be 
made.) 

* The proportions adopted by different builders, and by the same builder 
for different cases, being not precisely alike, the pupil is thus encouraged 
not to think any one set of given measurements indispensable. 



118 SUPPORTEKS AND CRANK MOTIONS. 

Ex. 2. A Standard for a Lathe. PL XVI., Fig. 1. 

Description. — This example illustrates the application of tan- 
gent lines and circles to the designing of open frames, having 
outlines conveniently varied for use and economy of material. 

The figure shows half of the side view (the object being sym- 
metrical), also an edgewise view. The scale, -J, being given, 
and the operations of construction being here more important 
than the precise measurements, only a few of the principal 
dimensions are given in this and in the following figures, leaving 
the rest to be assumed, or sufficiently determined by knowing 
the scale. 

The double lines on the edges indicate ribbed edges, so made 
to secure stiffness and strength. The central and triangular 
openings may also afford rests for long-handled tools or metal 
bars. The nut n secures the standard to the lathe-bed. 

Construction. — Having made the half widths 4^" and 12-J-" at 
top, and at AB, the outline BD may be drawn. This is com- 
posed of an arc of 60° with radius AB, a tangent to this, and a 
second arc of 60° tangent to the last line, and with its centre on 
a horizontal line through D. 

The outline of the central opening is partly concentric with 
the arc through D, partly circular with C as a centre, and 
partly circular as shown at the top, and there tangent to the side 
arcs. is here taken on a horizontal line through the lower 
end of the arc from D. 

Execution. — The figure, both halves of which should be 
drawn, and on a little larger scale, as -J- or -J-, gives occasion for 
the neat drawing of curved shade lines, and the neat connection 
of tangent outlines. 

Exercise. — 1. Vary the design by making the arcs from B and D each 
less than 60°, and so that C shall be on the radius through the lower 
limit of the arc through D. 

Ex. 3. Section of an Engine-bed, Guides, and their 
Support. PL XVI., Fig. 2. 

Description. — ABOD is a cross-section of the bed of a hori- 
zontal engine, which is of uniform section throughout, hh is a 
vertical plate, bolted to the bed as shown at n. From this 
plate, and solid with it, project two or more arms HH, which 



PLXY. 




SUPPORTERS AND CRANK MOTIONS. 119 

support the guide-bars GGr, between which slides the cross-head, 
not shown, to which the outer end of the piston-rod is fastened, 
as may be understood from the equivalent parts of nearly any 
locomotive or stationary engine. 

Construction. — Only the principal measurements being given, 
the others can be assumed, or made out by the given scale. The 
left side of the bed having vertical faces, these may be used as 
lines of reference from which to lay off horizontal measurements. 
Vertical ones can be laid off from the base line AB; or, on the 
guide attachments, from the top of the guides downward. To 
give greater stiffness to the arm H, its lower principal curve is 
struck from a centre, b, 1£" to the right of a, the centre of its 
semicircular outlines. The curved outlines generally are com- 
posed of circular arcs tangent to each other. 

As a minute following of given copies is not intended, these 
general explanations, measurements, and scale will sufficiently 
guide the learner in the construction of examples like the 
present. 

Execution. — The thin material of the bed gives occasion for 
section lines as fine and close together as can well be made. 

Exercises. — 1. Reverse the figure right for left. 

2. Supposing the guides to be four feet long, make a side and a plan 
view, showing three arms to the supporter H. 

7. Bearings. — This is a general term meaning any surface 
which immediately supports a moving piece. The bearings of a 
rotating piece are cylindrical and variously termed. Journals 
are formed in the frame of a machine and lined with brass or 
other anti-friction alloy. "When detached, they are pillow-blocks, 
as already shown. Bushes are whole hollow cylindrical linings 
of journals, but being unadjustable to compensate for wear, 
separate brasses are better. Foot-steps are the bearings at the 
base of vertical shafts, the lower end of which is a pivot. Axle 
boxes are the terminal supports of rail-car axles, and have a small 
vertical range of motion between the jaws of a stout iron frame. 

Cranks and Eccentrics. 

8. A crank, Fig. a, is an arm, CC, keyed at one end firmly 
to a revolving shaft SS' by a key kk' , and hence revolving 



120 



SUPPOETEES AND CEANK MOTIONS. 



with it ; and at the other end carrying a crank-pin pp', which is 
embraced loosely by a connecting rod. This connecting rod 
similarly embraces a parallel pin in the cross-head attached to a 
piston-rod, a pump-rod, or other piece having a reciprocating 
motion. Thus a rectilinear reciprocating motion,-as of a piston, 
is converted into a rotary motion, as seen in any locomotive, or 




Fig. a. 

a stationary engine of the usual type, or vice versa, as in case of 
a pump. The length of the stroke of the piston must evidently 
be equal to the diameter of the circle described by the centre of 
the crank-pin ; that is, equal to twice Sp. See Fig. b, where 
the stroke qq t of the forward eud of the connecting rod mr is 
equal to pp lf the dotted circle being that described by the centre 
of the crank-pin. 

9. Fig. b illustrates an important elementary point in crank 
motions. Eemembering that any connecting rod is of invariable 




length, take the middle-point m of the stroke qq x as a centre, and 



the length raS 



QP = QiPi 



of the rod as a radius, and the arc 



SUPPORTERS AID CKANK MOTIONS. 



121 



rSrj thus described will intersect the crank-pin circle in the 
corresponding positions r and r x of the crank-pin. 

Thus, while the cross-head pin passes oyer mq and qm, the 
crank-pin describes the arc r x pr, greater than a semicircle ; but 
while the former is passing over mq x and q r m, the crank-pin 
proceeds over rp x r lf less than a semicircle. 

Conversely, while the crank-pin traverses the rear semicircle 
ApB, the cross-head pin only travels from n to q and back ; but 
when the crank-pin describes the semicircle B/^A, the other 
pin travels from n to q x and back ; An being equal to r x m. 

"With the use of the connecting rod, this inequality mn, be- 
tween the two partial double strokes Qnq and Qnqi, would dis- 
appear only by using a rod of infinite length. But the em- 
ployment of a yoke with a slot, equal and parallel to AB, as in 



JB 



} 



v 






.S 



Fig. c f 



Fig. c, produces the same result by finite means. Here a piston- 
rod, P, issuing from the steam-cylinder C, is rigidly attached to 
a yoke, AB, in which the crank-pm 
plays as it is driven by the yoke. In this 
case the piston is exactly at the middle 
point of its stroke when the crank-pin 
is at either end of the diameter AB. 
This movement is often seen in steam 
fire-engines. 

10. Eccentrics. — The distance, Fig. a, 
from the centre of the shaft S to the 
centre of the crank-pin p, is called the 
arm of the crank. When this arm is so short, as compared with 
the diameter of the shaft, as to be entirely within the shaft, as 
at Sp, Fig. d, the crank-pin AB, whose centre is p, has to be 




Fig. d. 



122 



SUPPORTERS AND CRANK MOTIONS. 



made large enough to embrace the shaft. In this case the crank- 
pin is called an eccentric. 

That the eccentric is simply a short crank in principle and 
action, will be evident by substituting for the crank 0, Fig. a, 
a circular plate with centre p and radius sufficient to include 
the shaft. In either form of Fig. a, and in Fig. d, a connecting 
rod attached to the crank-pin would actuate any piece at its 
opposite end through a stroke equal to twice S^. 

11. A connecting rod is attached to a crank-pin by a method 
haying many modifications in minor details. The general prin- 
ciple, alike for all, is shown in Fig. e. The object to be secured 
is an invariable distance between the centres of the crank-pin 
and the pin p, at which the rod R is attached to the cross-head. 
E is the end of this rod, called the stub-end. ssss is the strap, 




Fig, e, 



in one piece. B, shown sectionally, and B' in elevation, are the 
brasses, square outside and cylindrical inside, which, together, 
embrace the shank of the crank-pin, and are kept from sliding 
off by the head of the crank-pin h, Fig. a. The whole is fas- 
tened by two bolts bb, bb. 

This arrangement being understood, suppose that by long 
wear the brasses play loosely upon the pin p. By driving in the 
slightly tapering key kk, its side aa presses the brass B against 
the pin p, the width of the slots ac ac in the strap permitting 
this to be done. Then loosening the bolts b, the holes for which 
are oblong from right to left, as seen in a plan view, a further 
driving in of the key kk operates through the hooked piece g, 
called a gib, to draw the strap s to the right, and thus draw up 
the brass B' against the pin p. 

Having understood one construction, the learner will be able 



SUPPORTERS AND CRANK MOTIONS. 



123 



to understand all the modifications which he may notice on loco- 
motive or other engines, such as the omission of the gib, which 
is unnecessary, with the bolts ; a separate key for each brass ; 
the stub-end extending to the left of kk, so as to wholly enclose 
it, when no bolts would be necessary ; a screw motion at the 
small end of h for drawing, instead of hammering in the key; etc. 

Ex. 4. A Crank. PL XVI., Fig. 3. This consists essen- 
tially of two collars connected by a tapering arm, the whole in 
one cast-iron piece. is the centre of the 8-inch shaft, and 
shaft collar of diameter ad, 19". P is the centre of the crank- 
pin, of 4", and of its collar, of 9" diameter. The arm CC is 
chambered as indicated by the dotted line around 0'. The 
linear arm OP is 24". 

The surfaces of the arm flow into those of the collars as indi- 
cated in line drawings by 



the curved 
upper edge 




ends of the 

of 0', lines 
whose geometrical con- 
struction is unnecessary in 
practice, but may be found 
as follows, Fig. /. In this 
figure the arc c'n' is that 
at en, Plate XVI., Fig. 3, 
enlarged, and the line On 
corresponds to PO. Then 
project points of c'n' upon 
en, as r' at r, and rr x and 
r'r" are the two projec- 
tions of the horizontal 
circle through rr', which 
cuts the edge n^o of the 
crank at r t , which, pro- 
jected upon the horizontal 
line r'r", gives r". Simi- 
larly, other points of the Jjl 9-J 
required curve n'n" r"o' are found. Such curves are, however, 
after the full-sized construction of a few cases, to apprehend their 
general form, sketched by hand, as they are not essential to a 
working drawing. . 



124 SUPPORTERS AS"D CRAS"K MOTIONS. 

Exercises — 1. Complete the crank, half of which is shown in the figure. 

2. Make a longitudinal section of the crank. 

3. Draw from measurement any accessible ribbed, trussed, or cham- 
bered crank. 

4. Draw a cranked axle (such as may be seen on old locomotives hav- 
ing "inside connections"). 

Ex. 5. A Ribbed Eccentric and Strap. PI. XVL, 

Figs. 4, 5. This may also be called an open or skeleton eccentric. 
is the centre, and Ob the radius of the shaft, 8-J-" diameter, 
to which the eccentric is clamped by clamp-screws, one of which, 
is n. The centre of the eccentric is a, which makes the crank- 
arm Oa of the eccentric 3", and hence the stroke, called the 
throw, of the valve, or whatever piece is moved by the eccen- 
tric, 6". 

The width of the different parts of the eccentric is shown on 
the fragment of sectional view, as at o'o" the thickness of the 
flange, or feather, o, the width e'e" of the collar b and rim e, 
and the width of the rib c. 

Fig. 5 shows a little more than one-quarter of the strap which 
surrounds the eccentric, and a little more than half of one of its 
two halves, which are bolted together through the ears, as cd. 
The arc ab is of the radius ae, Fig. 4 ; mn is of the radius ac, 
thus showing the groove in the strap which just fits the rib on 
the eccentric, and so prevents the strap from slipping off. The 
portion of the strap shown carries the socket ne, in which is 
keyed or clamped the eccentric rod, corresponding to the con- 
necting rod of a crank. The opposite or left-hand half is unin- 
terrupted in outline. The figure r'g' shows the form of the 
section at rg. 

Execution. — The numerous tangent arcs and curved heavy 
lines tapering at their termination will afford occasion for special 
care. 

Exercises. — 1. Draw the whole of the eccentric and its strap. 

2. Make a horizontal section of the eccentric. 

3. Make an end elevation of the eccentric and strap. 

Ex. 6. A Grooved Eccentric. PL XVI., Figs. 6, 7. 
This might also be called, by reason of its form, a chambered or 
box eccentric, since all of it between the solid collar Qa and the 



PL.XV1 




SUPPOKTEES AK"D CEAKK MOTIONS. 125 

rim cd consists essentially of two thin plates enclosing a hollow 
interior of width, 3f ", shown on the fragment of end elevation 
— the scale is -f%, as in Ex. 5. 

The shaft opening, of centre 0, is 6" in diameter, surrounded 
by solid metal 1" thick as indicated. The arm OQ being 4J", 
makes the throw of this eccentric 8J-". The opening, P, 5" 
diameter in the walls of the eccentric, gives access to the clamp- 
screw n by which it is fastened to the shaft. 

The strap, Fig. 7, a section of which is shown at H, sets in 
the groove cV of the circumference of the eccentric. Its outer 
arc mn is drawn from a centre a little to the right of that of AB, 
so as to support the rod-socket BO. As in the last example, the 
strap is in two halves bolted together through ears as at D. 

Exercises. — 1. Draw the whole eccentric with the strap in place upon 
it, and an end view of both. 

2. Make a horizontal and a vertical section (perpendicular to the 
paper) of the combined eccentric and strap. 

12. Chech, loch, or j 'ami) nuts. — On parts of machinery which 
are exposed to a jarring motion at high speed, as in locomotive 
machinery, two nuts are commonly seen at the 
end of the bolts which secure such pieces. These p— ^-' 

serve to clamp each other against the screw- =• 

threads of the bolt, and thus hold each other 
from working off the bolt. 

Other contrivances for securing the same re- 
sult, are nuts with notched sides, into which a 
detent enters, as may be observed in winding up F % V- 9> 

a watch; or a forked key kk' through the bolt and outside of 
the nut, as in Fig. g. 




CHAPTER II. 

GEAKLNG. 

13. Gearing is the term applied to wheels or straight bars 
when they are armed with interlocking teeth enabling them to 
take a firmer hold of each other, for the purpose of communi- 
cating motion, than they could if they were smooth surfaces, 
tangent to each other and communicating motion only by means 
of the friction of their surfaces of contact. 

In order to a smooth and uniform motion, the teeth must be 
equal and equidistant, and those on each body adapted to the 
form of those on the other body. Also, in order that the toothed 
bodies, two cylinders tangent to each other, for example, should 
preserve the distance between their centres, depressions beloiv 
the original surface of each body must be made between the teeth, 
in order to receive the portions which project beyond that of the 
other body. 

A toothed bar is called a rack; a toothed cylinder, a spur-ivheel, 
or pinion if small ; a toothed cone, a conical or bevel ivheel. 

14. Forms of teeth.— -1°. When a circle C, PL XVII., Fig. 1, 
rolls, without slipping, on a fixed straight line AB, any one point 
of the circle describes the curve called a cycloid. Thus the point 
describes the cycloid, one-half of which is OE', while the circle 
rolls to the position C. 

2°. When, on the contrary, a straight line as 3,3', Eig. 2, rolls, 
without slipping, on a fixed circle, any point of the rolling line 
describes the curve called an involute of the circle. Thus 3' de- 
scribes the involute 3', 2', 0. 

3°. Again: when one circle rolls on the exterior of another as 
the circle BB', Fig. 3, on the circumference BFA, any point on 
the rolling circumference traces the curve called an epicycloid. 
Thus the point F traces the half epicycloid FGr, while the circle 
C rolls to the position AG. 

4°. Finally: when BB', instead of rolling on the convex or ex- 
terior side of the circumference C, rolls upon its concave side, 






GEARING. 127 

or within it, any point of the rolling circle generates the cnrve 
called a hypocycloid. 
In all these cases the fixed line is called the base line or circle, 

15. Suppose now the circle BB', Fig. 3, to be revolved 180° 
about a tangent at B. It would then be tangent to the circle C 
interiorly at B, as it now is exteriorly at that point. 

If then, the diameter BB' being less than radius CB, the circle 
BB' rolls within C, on the arc BFA, the hypocycloid traced by 
B will be above AB. But if a circle of diameter greater than CB 
were to roll within on the arc BFA, the hypocycloid curve 
would be below AB. It plainly follows that the hypocycloid 
traced by the point B, when the circle of diameter just equal to 
CB rolls within C on the arc BFA, would coincide with BO. 
That is: the hypocycloid traced by any point of a circumference 
which rolls on the inside of a base circle of twice its diameter, 
is a straight line. 

16. These four curves (14) are suitable forms for the teeth of 
wheels, for the simple reason that when two circles as C and C, 
PI. XVII., Fig. 4, maintain rolling contact with each other, as 
at H, equal arcs of each come in contact in a given time. Hence 
the motion is the same in effect as if, separately, C rolled on C 
as a fixed circle, and then C rolled over an equal arc on C as a 
fixed circle, and therefore contact will be maintained by arming 
the wheels with teeth generated by the point H, for each case 
respectively. 

Similarly for a wheel C, and rack I'J\ 

The circle or line employed for generating the tooth curves is 
called their generating circle, or line. 

17. Construction of tooth curves. — This is very simple, and 
follows directly from the definitions in (14). Cycloid. Thus PI. 
XVII. , Fig. 1, the points 1, 2, 3, etc., on the circle C indicate 
the heights of above AB, corresponding to 1, 2, 3, etc., on AB 
as successive points of contact of C with AB. Then the inter- 
sections of the parallels to AB through 1, 2, 3, etc., on C, with 
the arcs of radii al, b2, c3, etc., will be points of the cycloid 0E'. 
Both sets of spaces 01, 12, etc., are equal, since there is no slipping 
of the circle in rolling on AB. Epicycloid. Likewise in Fig. 3, 
arcs Fl, etc., on circle C, = arc Fl, etc., on circle C, express 



128 GEARING. 

the character of the motion; a, b, c, etc., are positions of the 
centre C corresponding to 5, 4, 3, etc., as points of contact of 
the circles; the arcs of radii Cl, C2, C3, show the radial dis- 
tances of F from BFA as C rolls on 0; hence, -finally, the inter- 
sections, not lettered, of the arcs of radii el and Cl, d2 and C2, 
c3 and C3 are points of the epicycloids FG- and FB'. 

The liypocycloid is constructed in a precisely similar manner. 

The involute is approximately represented by tangent circular 
arcs as in Fig. 2. Here, 11', 22', 33', being positions of the roll- 
ing straight line at equidistant points of contact 1, 2, 3, we 
describe the arc 01' with radius 10 (taking the chord as approxi- 
mately equal to the arc), then an arc 1' 2' with radius 21', then 
the arc 2' 3' with radius 32', etc. The more numerous the 
points 1, 2, 3, etc., the closer will the compound curve thus 
found approximate to a true involute. 

These curves can be constructed mechanically on a large scale 
by means of a pin or pencil point inserted firmly in the edge of 
a wooden ruler or circle, either of which is made to roll without 
slipping on the other, or the circle on a fixed circle; or within a 
circular opening in a thin board, in the case of the liypocycloid. 

18. Definitions. — Let the circles of radii CH and C'H, PL 
XVII., Fig. 4, represent the original circumferences of two 
cylinders having IJ for a common tangent at H, and now pro- 
vided with interlocking teeth as shown, forming a pair of spur- 
wheels. These circles are called pitch-circles. The correspond- 
ing line I'J' of the rack is called its pitch-line. 

The distance db, or H'K on the rack, which includes a tooth 
and a space on the pitch-circle, is called the pitch. 

The circle of radius Cc is the root-circle, and contains the roots 
of the teeth. 

The circle of radius Qd is the point-circle, and contains the 
points of the teeth. 

The surfaces as bd are the faces of the teeth, and those as be 
are their flanks. 

19. Usucd proportions. — Supposing the pitch divided into 15 
equal parts, 7 of these are taken for the width, ah, of the tooth, 
leaving 8 of them for the width of the space, hb, to allow easy 
working of the teeth. Also 5£ of these spaces are taken for the 



GEARING. 129 

radial extent of the teeth beyond the pitch-circle and 6£ of them 
for their depth below the pitch-circle, to prevent the tooth points 
of one wheel from striking the rim of the other wheel. 

Application of Tooth Curves. 

20. Designing of gearing. — Comparing (15) and (16), the gen- 
erating circle of the tooth curves must be smaller than the pitch- 
circle in order to form the necessary flank surfaces (18). A 
common practice is, to employ for the flanks of each wheel a 
generating circle of diameter equal to the radius of the pitch-cir- 
cle of that wheel; in order to produce radial flanks, as most 
simple. Now, as seen by inspection of PL XVII., Fig. 4, the 
face of a tooth of each wheel is in contact with the flank of some 
tooth of the other wheel. Hence (16) the same circle that gen- 
erates the flanks of one wheel must generate the faces of the 
teeth of the other, since keeping the generating circle of diameter 
OH in contact with the pitch-circles at' their point of contact H, 
requires in effect the equal rolling of that generating circle upon 
the exterior of the circumference of wheel C, and on the interior 
side of that wheel 0. This can easily be seen experimentally by 
using three card-board circles. 

21. Detailed description. — Epicycloided teeth. — The circle CH 
generates the radial flanks, as he, of the teeth of C (14, 4°), and 
by rolling on the exterior of 0' generates the face curves of the 
teeth of C. To avoid confusion, HK may represent one of these 
curves, though it is really an involute. Likewise, the circle CH 
generates the radial flanks of wheel C, and by rolling on the 
exterior of win generate the epic}~cloidal faces of its teeth, 
found as in Fig. 3, but represented as before by the involute HL. 

22. Objections. — Each wheel having a separate generating 
circle, each will work correctly only with the other. But if one 
uniform generating circle be employed for the faces and flanks 
of any number of different-sized wheels of the same pitch, any 
two of them will work together properly. This common gen- 
erator must not exceed half the size of the least wheel of the set, 
so as to avoid convex flanks (15). 

23. Involute teeth.— Involute faces for both wheels can be 



130 GEARING. 

formed as shown by the rolling of the common tangent IJ at H, 
first on 0, giving the involute face curve HL (Fig. 4), and then 
on C, giving the face curve HK. 

Usually, however, when involute teeth are employed, they are 
not combined with radial flanks, since this violates the principle 
that the same generatrix should form the face and the flank 
which are to be in contact ; but IJ is made a common tangent 
through H to the root-circles of the two wheels, so that the in- 
volute teeth will be bounded by a single involute curve reaching 
to the root-circles, as they should, since a straight line cannot 
be rolled on the interior side of the pitch-circles to produce sepa- 
rate flank curves. 

24. The rack-generating circle continuing to be half the size 
of its own pitch-circle, the generating circle for the rack flanks 
will be I' J', since a straight line is a circle of infinite radius and 
half of that radius is infinite still. This understood, the flanks 
of the rack are straight lines as AH' perpendicular to its pitch- 
line, and the faces of the teeth of 0, being properly generated 
by the same line, are involutes as H'FT 

Likewise the flanks of the teeth of are straight lines genera- 
ted by the circle of diameter CH', while the faces of the rack 
teeth are cycloids as H'G' generated, as shown, by the rolling 
of the same circle on the pitch-line I' J'. But, as before, one 
generating circle can be used for faces and flanks of both 
wheels. 



Ex. 7. The Drawing of a Spur-wheel. — It is convenient 

that the pitch should be some simple measure as 1", 1£", 

li", . . . 2", . . . 2|", . . . etc., and it is necessary that 

the pitch should be contained an exact number of times in the 

pitch-circle. Hence the usual problem is: Given the pitch and 

number of teeth of a pair of wheels, to find their radii. 

Let P = pitclyNi = number of teeth, E = radius, 

and C = circumference of pitch-circle. 

Then C = P X N = 3.1416 X 2R, 

P X N 

whence R = - - — r— :, or denoting as usual, 3.1416 by n 

2 x 3.1416 ° 

R- PXN 



GEARING. 131 

Suppose a wheel of 24 teeth and \\" pitch. Its circumference 
will thus be 36" and its radius very nearly 5f ". 

The four quarters of the wheel being alike, it is sufficient to 
draw one of them, with the first tooth on the adjacent quarters, 
and this can conveniently be done on a scale of half the full 
size. 

Three forms of wheels are in use according to their size: solid 
wheels, as in PL XVII. , Fig. 4; plate wheels, consisting of a cen- 
tral hub, or doss, keyed to the shaft, and connected by a thin 
plate to the rim which carries the teeth; and armed ' wheels, in 
which the boss is connected with the rim by arms the perpen- 
dicular section of which is often an equally four-armed cross. 

Attending at first principally to the teeth, let the wheel now 
drawn be solid. 

Divide a quadrant of the pitch-circle carefully into six equal 
parts, one of which will be the pitch. 

Proportion the teeth by (19), giving the root and point circles. 
Lay off half the width of a tooth on each side of each point of 
division of the pitch-circle, which will make the lines as CH' 
and CD, Fig. 4, centre lines of teeth instead of as shown in 
that fio-ure. 

While teeth are shown in detailed working drawings, of full 
size and by the most accurate construction of their proper forms, 
they are approximately represented in general illustrative draw- 
ings, by various simple methods. Thus the faces may well be 
drawn by taking the pitch ab as a radius, with the centre, as at 
b, on the pitch-circle to draw the face aJ). A more summary 
process is shown in PI. XIX., Fig. 6. The flanks, if not radial, 
as shown in the figure, should be in reality hypocycloids (14, 15) 
which would diverge toivards the centre C, and which may suffi- 
ciently be represented by taking d, for example, for the centre of 
the flank beginning at n. 

Thus the elevation may be completed, placing the shade, or 
heavy lines, on each tooth by the usual rule, as shown. 

For the plan, draw two parallel lines at a distance apart equal 
to the width of the wheel; that is, the length of the teeth, which 
may be twice the pitch. Then simply project down the point 
angles as d, and visible root angles as c, and the points of con- 
tact of the face curves with tangents parallel to IJ, as at a and 



132 GEARING. 

near h. To become familiar with the subject, work out fully 
the following : 

Exercises— \. Construct the half, not shown, of the cycloid. PL 
XVII., Fig. 1. 

2. Complete both of the epicycloids half-shown in Fig, 3, one with the 
diameter of C equal to the radius of C. Also one, given by making the 
circles C and C equal. 

3. Construct the hypocycloid generated by the point H of circle C'H, 
Fig. 3, in rolling within circle C. 

4. Construct the hypocycloid generated by the circle CH 7 rolling 
within the small circle C. 

5. Construct an arc of the involute of circle C, generated by the point 
H' of the line I'J 7 , and by dividing a quadrant of C into eight equal 
parts. 

6. Draw a spur-wheel and rack, the wheel having 32 teeth and 2" 
pitch. Make the drawing of full size, showing a quadrant only of the 
wheel, bisected at its point of contact with the rack, and let the faces 
and flanks of both pieces have one generating circle whose diameter shall 
be f that of the radius of the wheel. 

7. In Ex. 6, substitute for the rack a wheel of 20 teeth, and let the 
common generating circle of the teeth-profiles of both wheels be of less 
diameter than the radius of the smaller wheel. 

8. Draw enough of a four-armed wheel of 30 teeth and 1^" pitch to 
show two arms fully, making the thickness of the rim, and of the arms, 
and of the feather, and their width also (see o, PI. XVI., Fig. 4), 
which surrounds the openings between the arms, all equal T 7 -g- of the 
pitch, and the radial thickness of the hub |- of the pitch. 

9. Draw PI. XIX., Fig. 6, twice its present size or larger, and first 
with involute teeth, and then with epicycloidal faces and hypocycloidal 
flanks, and after constructing carefully one tooth-profile, find by trial 
the centre and radius of the circular arc which will most nearly coincide 
with it, to use in drawing the other teeth. 

25. Velocities. — It is clear (13) that if one wheel has 30 teeth 
and another 60, the former must make two revolutions to one of 
the latter, also that the radius of the former is one half that of 
the latter. What is true for one such case is evidently true in 
principle for all cases. That is, the number of revolutions in a 
given time of each of a pair of toothed wheels is inversely as its 
number of teeth, or as its radius. 

The circumference velocities, as at the point of contact H, 
PI. XVII., Fig. 4, are necessarily equal, but the velocities at the 



GEARING. 133 

same distance from the centre, as 1 foot, on both wheels are as 
the numbers of revolutions, and hence inversely as their radii. 
The latter are termed angular velocities. Then denoting them 
by V and v for the wheels C and C respectively, and the radii 
CH by R and C'H by r, we have 

V : v : : r : R. 

Bevel and Mitre WJieels. 

26. PL XVIII. , Fig. 1, shows a pair of bevel wheels. These 
consist of a pair of frusta of cones, CAD and one of which CAB 
is the half, provided with teeth which converge to the common 
vertex, C, of the cones, whose axes, CB and CF, may make any 
angle with each other. 

When, as in Fig. 2, the axes are at right angles, the wheels 
are distinguished as mitre tvheels. 

As is lowered nearer and nearer' to AB, still continuing the 
common vertex of the cones, the wheel AB becomes natter and 
flatter, and when finally C passes below AB, the wheel AB be- 
comes a hollow frustum toothed on its inner surface. 

On account of the intersection of the axes of bevel wheels, 
one or both of the axes terminate at the wheels, as in Figs. 1 
and 2. 

27. Velocities. — The principles of (25) apply to bevel wheels. 
Hence having given one wheel, as CxiB, Fig. 1, and the ratio 
of the velocities, make a'd' and a'e' in this ratio, a'cV repre- 
senting the relative velocity of the required wheel, and Ce' will 
be the axis tangent from C to an arc of centre a' and radius a'e' 9 
and AD the diameter of the latter wheel. 

Or, having given the vertex C, and axes CB and CF, set off 
(V and Ob' inversely as the two velocities (that is, set off on 
each axis a distance proportional to the velocity of the other 
axis), and complete the parallelogram Cb'a'c', and CA is the 
line which will divide the angle BCF included by the axes, so 
as to give the radii AB and -|AD of the required wheels. 

When, as in Fig. 2, the axes are at right angles, the latter 
construction applies, but the parallelogram becomes the rect- 
angle CbaO. 



134 GEARING. 

Ex. 8. To Draw a Pair of Bevel Wheels. PL XVIII., 

Figs. 2-5. 

Let the cones CAB and CAD — called the pitch-cones, because 
they contain the pitch-circles — be given. At A,- the point of 
contact of the pitch-circles, draw EAF perpendicular to CA, and 
draw EA, EB, FA, FD. Then EAB and FAD will be the 
cones containing the larger, or outer ends of the teeth. Next, 
laying off AI equal to the length of a tooth, and drawing IE par- 
allel to AD, IH parallel to AB, and GIJ parallel to EF, we have 
JIR and GIH, the cones containing the inner ends of the teeth. 

The wheels here shown have respectively 36 and 28 teeth. 
Then divide each quadrant of the semi-pitch-circle on A'B' into 
9 nine equal parts, and each quadrant of the semi-pitch-circle on 
A"D' into 7 equal parts. Taking the proportions before used, 
make Be and BZ>, each on BE, respectively 5^ and 6| fifteenths 
of the pitch, to obtain the point and root circles parallel to AB 
through e and b, since the real height be of the teeth is shown 
in its real size on the extreme element EB of the cone EAB. 
The horizontal projections of these circles are those with radii 
CV and G'V. 

The corresponding inner point and root circles are found by 
noting g, the intersection of eC and GH, and that of bC with 
GH. This last point is horizontally projected at n' . 

Having thus both projections of all the circles of construc- 
tion: 

1°. Lay off ^ of the pitch, that is, half the space between 
two teeth, on each side of A', B' and S, and from the points so 
found lay off the pitch, over and over, which will give all those 
points of the teeth which are in the outer pitch-circle A'SB'; 
and project these points on AB. 

2°. Through the points just found on A'SB' draw lines to C, 
limited by the circle CV; and through those on AB, lines to E 
limited by ab, for the outer ends of the flanks. 

3°. From the points on ab draw lines to C, limited by the 
vertical projection of circle C V, for the root lines of the teeth, 
and from the points thus found, the inner ends of the flanks 
radiating from G and limited by HI. 

4°. Make arcs, tangent to each other as at 0, Fig. 5, with 
radii EA and FA, which will be (Div. I., Prob. 28) arcs of 



K 








V n" - 








-i 

1 


T?X 




"1 






-ii* 


\ 


j. 

I 
1 





— 


-- 


_$\ 


_L 










—•t 



PL^VII. 




GEAKING. 135 

the developments of the outer pitch-circles of the two wheels. 
On these lay off the pitch, and proportion the teeth as already 
described, the flanks running to E (below the border) and F, 
and the faces drawn with convenient circular arcs to replace the 
epicycloidal curves OP and OQ. Having thus found the width 
of the teeth at their outer points, lay off half this width on 
each side of the middle point of each tooth on the circle of 
radius CV. 

5°. Through these points on circle CV draw lines to C, lim- 
ited by circle C'g' ; project the points of circle CV upon ec, 
vertical projection of circle C V, and thence draw the point edges 
of the teeth towards C. 

6°. Finally, the face curves at both ends of the teeth are 
sketched by hand, tangent to the flanks. 

By precisely similar operations, the two projections of the 
wheel AD — A"D' may be drawn. 

The hub and arms of both can be easily drawn, as shown. 

To become perfectly familiar with the operations here de- 
scribed, work out the following variations : 

Exercises. — 1. Changing the numbers of teeth, let the axis of the wheel 
AD be perpendicular to the paper at C, so as to appear as Fig. 4 now 
* does. 

2. Again changing the number of teeth, let the wheel AD be in gear 
with AB at BH, and then draw the figure as if PI. XVIII. were upside 
down, making CA'SB' the elevation, instead of, as now, the plan. 

Screivs and Serpentines. 

28. Triangular -threaded screivs. — If the isosceles triangle 
cdl2, PI. XIX., Fig. 1, whose base is in the vertical line E/j, be 
revolved, together with that line, uniformly, around the vertical 
AB as an axis, having also a uniform vertical motion on Wi, it 
will generate the spiral solid called the thread of a triangular- 
threaded, also called a V-threaded screw. The surfaces gen- 
erated by dl2 and cl2 are helicoids, upper and lower. The lines 
generated by the points c, d and 12 are helices, inner and outer. 
~Eh will generate a cylinder, called the eore or newel of the 
screw. 

29. Square-threaded and other screws. — If, PI. XIX., Fig. 2, 



136 GEARIXG. 

a square, EC 6, be substituted for the triangle, the result will be 
a square-threaded screw. 

If, Fig. 3, a sphere whose centre describes a helix be the gen- 
eratrix, the resulting solid will be that called a serpentine. This 
is the form of a spiral spring formed of circular wire ; also of the 
hand-rail of circular stairs, when the rail has a circular section 
made by cutting it " square across." 

Again : if, Fig. 5, the profile of a tooth be taken as the gen- 
eratrix of the thread, there will be formed the kind of toothed 
wheel called an endless screw, since its constant rotation in one 
direction will actuate the wheel L. It is always the screw that 
is the "driver" and actuates the wheel, which is the "follower" 
and receives a very slow motion ; since the tooth G will be car- 
ried to the position of the next tooth above it, by one complete 
revolution of the screw. 

30. Number of threads. — In PL XIX., Fig. 1, one helical revo- 
lution of the generating triangle brings the side 6*12 to the posi- 
tion d24r, which allows no intermediate position of the triangle. 
The screw is therefore single -threaded. The like is true of the 
screws in Figs. 2 and 5. 

If, however, Fig. 1, one such revolution of cdl2 had, by 
means of a greater ascending motion, brought cl2 to the. position 
rs, the screw would have been hvo-threaded; and if to the posi- 
tion no, it would have been three-threaded. The like again is 
true of other screws, the number of threads being adapted to 
the advance parallel to AB, of any point of the screw in one 
revolution. This advance is called the pitch of the screw. 

Evidently the coils of a second spiral like Fig. 3 could be laid 
between those shown. It would then be two-threaded. 

Ex. 9. To Construct the Projections of a Triangu- 
lar-threaded Screw. PI. XIX., Fig. 1. 

The construction of the screw consists principally in that of its 
helices. Accordingly, let AE and AC be the radii of the circles 
which represent the circular motions of the points c and 12, and 
which are the horizontal projections of the inner and outer 
helices. And let 0,12 be the pitch of the screw. 

As both component motions, circular and rectilinear, of the 



GEAKING. 137 

compound helical motion are uniform, divide the circles AE and 
AC and the pitch 0,12 all into the same number of equal parts, 
here 12, and draw horizontal lines through the points of division 
on 0,12. Then for an outer helix, project C at 0; 1 on the first 
horizontal above it ; 2 on the second horizontal, 6, at 6 on the 
sixth horizontal, and so on till C is projected again at 12. 

Proceed in a precisely similar manner, beginning by project- 
ing E at c, to find points of an inner helix. The lines as cl2 
and c?12 complete the figure. 

Each half, to the right and left of AB, of the visible front 
half, as 06, of an outer helix is like the other half reversed, both 
right for left and upside down. Hence, as all the outer helices are 
alike, the portion of an irregular curve which will fit one half of 
one, will serve in ruling them all. Similar remarks apply to the 
inner helices. 

Had the ascent been from D to the left on the front half of 
the screw instead of from C to the right, the screw would have 
been left-handed. Left-handed screws are only employed for 
special purposes, as when two rods, placed end to end, are to 
be separated or brought together by a screw link working on 
both, as seen in the truss-rods under rail-car bodies. In this 
case the screw-threads on one rod would be right-handed, and 
those on the other left-handed. 

Exercises. — 1. Construct the projections of a two-threaded and of a 
three-threaded triangular screw. 

2. Construct the projections of a two-threaded and of a three-threaded 
left-handed screw. 

Ex. 10. To Draw a Square-threaded Screw. PI. 
XIX., Fig. 2. 

The operations in this case are so similar to those of the 
last problem, as is evident from the figure, that they need no de- 
tailed description. The form of the thread renders the under 
outer helices of the left side, and the upper outer helices of 
the right side, of the screw visible on the back half of the screw 
until they disappear behind the cylindrical core. Also, the inner 
helices are visible only on the under left-hand side and upper 
right-hand side of the thread. 

In the execution, it is very important to remember that any 



138 GEARING. 

one helix is, on the screw itself, of uniform curvature through- 
out, hence though very sharply curved in projection at the 
extreme points, as 6 and 12, especially in a single-threaded 
screw, they are not there pointed, except in drawings on a small 
scale where they may be approximately represented by straight 
lines, as in Figs. 7, 9, and 10. 

Exercise. — Draw a square-threaded screw with three threads, and 
show all four helices of one thread throughout, but dotted where 
invisible. 

Ex. 11. To Draw the Interior of a Nut or Internal 
Screw. 

PL XIX., Fig. 8, shows the interior of one half of the nut 
for a square-threaded screw ; that is, of the hollow cylinder with 
a thread on its interior surface, adapted to work in the spaces 
between the threads of the screw. The figure representing the 
rear half of the nut, the threads must there ascend to the left, 
as they do on the rear half of the screw. 

Exercises. — 1. Draw the vertical section of the nut corresponding with 
Fig. 1. 

2. Draw that of the nut of a square-threaded screw of two threads. 

Ex. 12. To Draw the Endless Screw and Worm 
Wheel. PL XIX., Figs. 4, 5. 

The profile of a tooth here becomes the generatrix of a screw- 
thread bounded by helices found as before. The pitch-line MX 
is divided by the pitch as m the case of a rack, the pitch of the 
screw and wheel being the same. 

The wheel, having its axis in a direction perpendicular to that 
of the screw, is in reality a short piece of a screw having a very 
great pitch. That is, the angle made by the helices of the 
wheel-teeth with a plane perpendicular to the axis of the wheel, 
that is, with the plane of the paper, is the complement of the 
angle made by the screw helices with a plane perpendicular to 
its axis, that is, to a plane perpendicular to the paper on GD. 
The curves, as that to the left of X, which represent the further 
ends of the teeth, are assumed, unless the width of the wheel is 
shown by a plan view. 



GEARING. 139 

Ex. 13. To Draw a Serpentine. PI. XIX., Fig. 3. 

This surface is one which, like a thin helical tube, would in- 
close, tangentially, all the positions of a sphere, indicated by 
the dotted circles, whose centre should describe a helix, ACB — 
2345. 

The contours, or apparent bounding lines, of the serpentine are 
not helices, though at a uniform perpendicular distance from the 
central helix, but are drawn tangent to the numerous equal 
dotted circles haying their centres on the helix, and which repre- 
sent as many positions of the generating sphere. 

Surfaces which, like the sphere and serpentine, are nowhere 
straight, are call double-curved Where partly convex, as on the 
outer circle, or in the circle OF, and partly concave, as on the 
inner side, or on the circle OD, the contour vanishes into the 
surfaces, at certain points, when shown by a line drawing, as is 
seen at the left of the under contours, and the right of the 
upper ones. 

The lower coil is shown approximately as straight, indicating 
#hat would be permissible in rough drawings or on a small scale. 



DIVISION SIXTH. 

SIMPLE STRUCTURES AND MACHINES. 



237. Note. The objects of this Division are, to acquaint the 
student with a few things respecting the drawing of whole structures 
which are not met with in the drawing of mere details ; to serve as 
a sort of review of practice in certain processes of execution ; and 
to afford illustrations of parts of structures whose names have yet 
to be defined. Proceeding with the same order as regards material 
that was observed in Division Second, we have : — 



CHAPTER I. 

STONE STRUCTURES. 

238. Example 1°. Abriok segmental Arch. PL XX., Fig. 123. 

Description of the structure. — A segmental arch is one whose 
curved edges, as aCc, are less than semicircles. A brick segmental 
arch is usually built with the widths of the bricks placed radially, 
since, as the bricks are rectangular, the mortar is disposed between 
them in a wedge form iu order that each brick with the mortal 
attached may act as a wedge ; while if the length of the bricks be 
radial, tne mortar spaces will be inconveniently wide at their outer 
ends, unless the arch be a very wide one, or unless it have a very 
large radius. 

The permanent supports of the arch, as wPT, are called abut 
ments, and the radial surface, as ?iab, against which the arch rests, 
is called a skew-back. 

The temporary supports of an arch while it is being built are 
called centres or centrings, and vary from a mere curved frame 
made of pieces of board — as used in case of a small drain or round 



PL.XIX 




STONE STRUCTURES. J 41 

topped window — to a heavy and complicated framing, as used foi 
the temporary support of heavy stone bridges. 

Note. The general designing of these massive centrings may 
call for as much of scientific engineering knowledge, and their details 
and management may call for as much practical engineering skill, 
as does the construction of the permanent works to which these 
centrings are auxiliary. In short, the detailed design and manage- 
ment of auxiliary constructions, in general, is no unimportant depart- 
ment of engineering study. 

The span is the distance, as ac, between the points of support, on 
the under surface of the arch. The stones over the arch and abut- 
ment, form the spandril, or backing, Qc£P. 

239. Graphical construction. — Let the scale be one of four feet 
to the inch = 48 inches to one inch — T \. Draw RT to represent 
the horizontal surface on which the arch rests. Let the radius of 
the inner curve of the arch be 7 feet, the height of the line ac from 
the ground 2 feet 8 inches, and the span 7 feet. Then at some 
point of the ground line, draw a vertical line, OC, for a centre 
line ; then draw the abutments at equal distances on each side of 
the centre line, and 6 feet 8 inches apart. Let them be 2 feet 6 
inches wide. 

Since the span and radius have been made equal, Ob and O d may 
be drawn, in this example, with the 60° triangle. Drawing these 
lines, and making Oa—1 feet, make ab = o\\Q foot, draw the two 
curves at the end of the arch, and make b and d points in the top 
surfaces of the abutments. 

To locate the bricks, since the thickness of the mortar between 
the bricks, at the inner curve of the arch, would be very slight, lay 
off two inches on the arc aCc an exact number of times. The dis- 
tance taken in the compasses as two inches, may be so adapted as 
to be contained an exact number of times in aCc, since the thick- 
ness of the mortar has been neglected, but would in practice be so 
adjusted, as to allow an exact number of whole bricks in each 
course. 

The arch being a foot thick, there will be three rows of bricka 
seen in its front. Draw therefore two arcs, dividing ab and cd into 
three spaces of four inches each, and repeat the process of division 
on both of them. 

Having all the above-named divisions complete, fasten a fine 
needle vertically at O, and, keeping the edge of the ruler against it, 
to keep that edge on the centre without difficulty, draw the lines 
which represent the joints in each of the three courses of brick. 



14:2 STONE STRUCTURES. 

240. Ex. 2°. A semi-cylindrical Culvert, having vertical 
quarter-cylindrical Wing Walls, truncated obliquely. PI, 
XX., Fig. 124. 

Description of the structure. — A culvert is an arched passage, 
often flat bottomed, constructed for the purpose of carrying water 
under a canal or other thoroughfare. Wing walls are curved con- 
tinuations of the vertical flat wall in which the end of the arch is 
seen. Their use is to support the embankment through which the 
culvert is made to pass, and to prevent loose materials from the 
embankment from working their way or being washed into the cul- 
vert. Partly, perhaps, for appearance's sake, the slope of the plane 
which truncates the flat arch-wall, called the spandril wall, and the 
wing walls, is parallel to the slope of the embankment. The wing 
walls are often terminated by rectangular flat-topped posts — " piers" 
or "buttresses," AA', and the tops, both of these piers and of the 
walls, are covered with thin stones, abed — a"b"c"d\ broader than 
the wall is thick, and collectively called the coping. 

Since the parts of stone structures are not usually firmly bound 
or framed together, each course cannot be regarded as one solid 
piece, but rather each stone, in case, for instance, of the lowermost 
course, rests directly on the ground, independently of other stones 
of the same course, hence if the ground were softer in some spots, 
under such a course, than in others, the stone resting on that spot 
would settle more than others, causing, in time, a general disloca- 
tion of the structure. Hence it is important to have what are called 
continuous bearings, that is, virtually, a single solid piece of some 
material on which several stones may rest, and placed between 
the lowest course and the ground. 

Timbers buried away from the air are nearly imperishable ; hence, 
timbers laid upon the ground, if that be firm, and covered with a 
double floor of plank, form a good foundation for stone structures; 
and in the case of a culvert, if such a flooring is made continuous 
over the whole space covered by the arch, it will prevent the flow- 
ing water from washing out the earth under the sides of the arch. 

When the wing walls and spandril are built in courses of uni- 
form thickness, the arrangement of the stones forming the arch, 
so as to bond neatly with those of the walls, offers some difficul- 
ties, as several things are to be harmonized. Thus, the arch stones 
must be of equal thickness, at least all except the top one, and then, 
there must be but little difference between the widths of the top, 
or hey stone, and the other stones; the stones must not be dis- 
proportionately thin or very wide, they should have no re-entrant 



STONE STRUCTURES. 143 

angles, or very acute angles, and there must not be any great 
extent of unbroken joint. 

241 . Graphical constt^uction. — Let the scale be that of five feet 
to an inch = 60 inches to an inch =3*5-, 

a. Draw a centre line, BB', for the plan. 

b. Supposing the radius of the outer surface, or back, of the arch 
to be 5j feet, draw CC parallel to BB' and 5|- feet from it. 

c. Draw BE, and on C'C produced, make EC = 9 feet 8 inches, 
CD, the thickness of the face wall of the arch =2 feet 4 inches, and 
the radius, oD, of the face of the wing wall=4 feet. 

d. With o as a centre, draw the quadrants CG, and DF, and with 
a radius of 3 feet 8 inches, draw the arc cA, the plan of the inner 
edge of the coping. Also draw at D and C, lines perpendicular 
to BB' to represent the face wall of the arch. 

e. At G, draw GA towards o, and =3 feet, for the length of 
the cap stone of the buttress, AA', and make its width = 2 feet 
10 inches, tangent to CG at G. The top of this cap stone, being a 
flat quadrangular pyramid, draw diagonals through G and A, to 
represent its slanting edges. 

f. Supposing the arch to be 1| feet thick, make C'H=1^ feet, 
and at C and H, draw the irregular curved lines of the broken end 
of the arch, and the broken line near the centre line, also a fragment 
of the straight part of the coping. 

g. Let the horizontal course on which the arch rests, be 2 feet 9 
inches wide, i. e., make He=:3 inches, and CVi=l foot ; and let the 
planking project 3 inches beyond the said course, making er—3 feet. 
Through e, n and r, draw lines parallel to BB' and extending a little 
to the right of C'H. 

A. Proceeding to represent the parts of the arch substantially in 
the order of their distance from the eye, as seen in a plan view, a 
portion of the planking may next be represented. The pairs of 
broken edges, and the position of the joints, show that there are 
two layers of plank and that they break joints. 

i. Under these planks, appear the foundation timbers, which 
being laid transversely, and being one foot wide and one foot apart, 
are represented by parallels one foot apart, and perpendicular to 
BB'. Let the planking project 4 inches beyond the left hand tim 
ber. Observe that two timbers touch each other under the arch 
front. 

j. The general arrangement of stones in the curved courses of the 
wing wall, in order that they may break joints, is, to have three and 
four stones, respectively, in the consecutive courses. To indicate 



144 



STOXE STRUCTURES. 



this arrangement in the plan, hQ,fb, gd and DC will represent tne 
joints of alternate courses, and the lines km, &c. midway between 
the former, will represent the joints of the remaining intermediate 
courses. 

This completes a partial and dissected plan which shows more of 
the construction than would a plan view of the finished culvert, and 
as much, as if the parts on both sides of the centre line were shown. 
In fact, in drawings which are strictly working drawings, each pro- 
jection should show as much as possible in regard to each distinct 
part of the object represented. 

242. Passing to the side elevation, which is a sectional one, show- 
ing parts in and beyond a vertical plane through the axis of the 
arch, we have : — 

a. The foundation timbers, as m'q, &c, projected up from the 
plan ; or, one of them being so projected, the others may be con- 
structed, independently of the plan, by the given measurements. 

b. The double course of planking op, appears next with an occa- 
sional vertical joint, showing where a plank ends. 

c. The buttress, A, and its cap stone Y, are projected up from 
the plan, and made 6 feet high, from the planking to G'. 

d. From G' and h\ the slanting top of the wing walls are shown, 
as having a slope of 1^- to 1 — i. e. h'ti' '=% h"u — and the vertical lines 
at C, D' and D" are projected up from C, D and D'". 

The remaining lines of the side elevation are best projected back 
from the end elevation, when that shall have been drawn. 

243. In the end or front elevation, we have : — 

a. At m"rn'" , a side view of one of the foundation timbers, 
broken at m'", so as to show other timbers behind it. 

b. The planking o'o" in this view, shows the ends of the planks 
in both layers — breaking joints. 

c. T$o' = Bo'", taken from the plan ; and in general, all the hori- 
zontal distances on this elevation, are taken from the plan, on lines 
perpendicular to BB'. 

d. The vertical sides of the buttress, A', are thus found. The 
heights of its parts are projected over from the side elevation. 

e. The thickness of the foundation course, ts = l? feet, and tr'=en, 
on the plan. 

/. The centre, O, of the face of the arch, is in the line r't pro- 
duced. The radius of the inner curve (intrados) of the arch is 4 feet 
and of the cylindrical back, behind the face wall, 54- feet — shown by 
a dotted arc. In representing the stones forming the arch, it is to 
be remembered that they must be equal, except the " key stone," 



STONE STRL'dREfe. 



145 



stances, o'"b"\ &c, from o"\ 
o'N, as at o'b'"\ &c, and at 
front elevation are drawn in 



g, which may be a little thicker thaiie others; they must also be 
of agreeable proportions, free fromjry acute angles, or from re- 
entrant obtuse angles ; and must infere as little as possible with 
the bond of the regular horizontal cp'ses of the wing walls. There 
must also be an odd number of sto^ (ring stones) in the front of 
the arch. 

On both elevations, draw the Hzontal lines representing the 
wing wall courses as one foot in ickness, and divide the innei 
curve of the arch into 15 equal j-s. Draw radial lines through 
the points of division. Their intentions with the horizontal lines 
are managed according to the prirles just laid down. 

g. The points, as k andy, in them, are then projected into the 
alternate courses of the side elevm, and into the line, Bo"', of 
the plan. 

From the latter line, the severa 
thus found, are transferred to the 
these points the vertical joints of 
their proper position, as being theme actual joints, shown by the 
vertical lines of the side elevation, i the stones immediately under 
the coping, there must generally some irregularity, in order to 
avoid triangular stones, or stones nappropriate size. 

h. To construct the front eleva i of the coping. All points, a& 
a, a', a", in either the front or bapr upper or lower edges of the 
coping, are found in the same wand as follows : 

a" is in a horizontal line throug' 
distance from o' equals the distam 
ing other points similarly, the ed 
with an "irregular curve." 

The horizontal portion of the (ng, over the arch, is projected 
over from C and from the two ejof the vertical line at D'. 

Execution. — In respect to this, 

Example. Let tins design, oi 
scale of four fe?t to the inch, on j 
place the three projections in 
shown (15) and (32). 



"a"' on the plan. Construct- 
of the coping may be drawn 



drawing explains itself. 
y similar one, be drawn on a 
ger plate; not forgetting to 
proper relative position, as 



CHPTER II. 

WOOD] STBUCTURES. 

244. Ex. 3°. Elevation a " King Post Truss." 
Mechanical construction, d— A Truss is an assemblage of pieces 

so fastened together as to be*tually a single piece, and therefore 
exerting only a vertical forceue to its weight, upon the support- 
ing walls. 

In PI. XX., Fig. 125, A itie beam; B is & principal; C is a 
rafter ; D is the king post ; s a strut ; F is a wall plate ; G is a 
purlin — running parallel to thidge of the roof, from truss to truss, 
and supporting the rafters, is the ridge pole ; W is the wall^ 
and ab is a strap by which the beam is suspended from the king 
post. 

245. Graphical constructit—bi the figure, only half of the truss 
is shown, but the directions ay to the drawing of the whole. In 
these directions an accent, tt ' , indicates feet, and two accents, 
" , inches. For practice draw whole figure, and on a larger scale. 

a. Draw the vertical centme bD. 

b. Draw the upper and \o\ edges of the tie beam, one foot 
apart, and 12' in length, on e side of the vertical line. 

c. On the centre line, lay ffora the top of the tie beam, 5' — 6' 
to locate the intersection of tops of the principals ; and on the 
top of the tie beam, lay off bn each side, to locate the intersec- 
tion of the upper faces of the icipals with the top of the tie beam. 

d. Draw the line joining itwo points just found, and on any 
perpendicular to it, asfg, lafits depth = 8", and draw its lower 
edge parallel to the upper ec Make the shoulder at o=3" and 
parallel to fg. 

e. From the top of the b, draw short indefinite lines, <?, 6 f 
each side of the centre line, note the points, as e y where they 
would meet the upper sides he principals. 

f. Draw vertical lines OAch side of the centre line and 4* 
from it. 

g. From the points, as eaw lines parallel to fg till they inter- 
sect the last named verticaes. 



WOODEN STRfUKES. 



147 



h. Make ns=5' — 9*. Make the j>rt vertical distance at c=4' 
draw sc, and make the upper side (foe strut parallel to sc, and 4 
from it. Note the intersection of is parallel with the line to the 
left of D, and connect this point 1th the upper end of c, to com' 
plete the strut. 

i. Draw the edges of the rafter, jjrallel to those of the principal, 
4' apart, and leaving 4" between h rafter and the principal. At 
o, draw a vertical line till it mee ;he lower edge of C, and from 
this intersection draw a horizonta lie till it meets the upper edge 
of C ; which gives proper dimensi 's to the wall plate. 

j. From the intersections of th ipper edges of the rafters, lay 
off downwards on the centre linJ^-2% and make the ridge pole, 
thus located, 3" wide. 

k. In the middle of the upper «ge of the principal, place the 
purlin 4* X 6", and setting 2" into tit principal. 

I Let the strap, ab, be 2" wide, Id 2'— 6" long from the bottom 
of the tie beam. Let it be spiked jb the king post and tie beam, 
and let it be half an inch thick, as shown below the beam. W, the 
supporting wall, is made at pleasui^. 

Execution. — This mainly explain itself. As working drawings 
usually have the dimensions figured upon them, let the dimensions 
be recorded in small hair line figmes, between arrow heads which 
denote what points the measuremeDts refer to. 

246. Ex. 4°. A "Queen Post Truss" Bridge. PI. XXL, 
Fig. 126. 

Mechanical construction. — This* j a bridge of 33 feet span, ovei 
a canal 20'— 6" wide between its 1 anks at top, and 20' — 2" at the 
water line. It rests on stone abuti ients, R and P, one of which is 
represented as resting on a plank a id timber foundation, the other 
on " piles." 

A is the tie beam ; B, B' the qvt'en posts ; C, C the principals; 
D the collar beam, or straining sill } ' R, P, the abutments ; eQ,t the 
pavement of the tow path ; iK th| 
TT the opposite timber wall, held 
into the wall timbers ; E, S, the pile 
are the principal parts. 

247. Graphical construction, — It 
to the inch. 

a. All parts of the truss are laid clff on, or from, the centre line 
AD. A is 14 ff deep; the dimension} 
top, where they are 10' wide for a 



stone side walls of the canal; 
by timbers UU', N, dovetailed 
s, iron shod at bottom. These 

t the scale be one of five feet 



j of BB' are 12' x 6', except at 
ertical space of 16'. C and D 



148 



WOOIX STKUCTURES. 



are each 10" deep. BB' are Y apart, and the feet of C and C, 12* 
from the ends of the tie bea, which is 36' long. D is 6* below 
the top of the queen posts, rare inch rods with five inch washers, 
*" thick, and nuts2£"xl". b' is a £ " bolt; with washer 4"x|' 
and nuts, 2'xl"; and perpen-cular to the joint, ad. 

b. From each end of the tie earn, lay off 1' — 9" each way for the 
width of the abutments, at the )p. Make the right hand abutment 
rectangular in section and 11' gh, of rectangular stones in irregu- 
lar bond (76). Let the left har. abutment have a batter of 1" in 1' 
on the side towards the cana and let it be eleven feet high, in 
eleven equal courses. 

c. Make et, the width of the aved tow path =7' — 6% with a rise 
in the centre, at Q, of Q". 

d. The side wall is of rubble,!:' thick at bottom, and extending 
18* below the water, with a bater of 1" in 1', and having its upper 
edge formed of a timber 12" sqire. 

e. The right hand abutment r;ts on a double course of three-inch 
planks, qq\ b^' broad, and resng on four rows of 10" piles, ES. 
S is the sheet iron conical shoe it the lower end of one of these 
piles, the dots at the upper erl of which represent nails which 
fasten it to the pile. 

f. TT is a timber wall havinga batter of V to 1', and held in 
place by timbers, UU', N, dovetiled into it at its horizontal joints, 
in various places. 

g. The water line is 2' below '^, and the water is 4^ feet deep. 
248. Execution. — It is intend'! that this plate should be tinted, 

though, on account of the dimci J :y of procuring adequate engraved 
fac-similes of tinted hand-made drawings, it is here shown only aa 
a finished line drawing, and as s ch, explains itself, after observing 
that as the left hand abutment shown in elevation, it is dotted 
below the ground ; while, as th right hand abutment is shown in 
section, it is made wholly in full' lines, and earth is shown only at 
each side of it. 

The usual conventional rule is, to fill the sectional elevation of a 
atone wall with wavy lines; bu;Uvhere other marks serve to distin- 
guish elevations from sections., as in the case just described, this 
labor is unnecessary. 

The following would be the general order of operations, in case 
this drawing were shaded. 

a. Pencil all parts in fine fair*, lines. 

b. Ink all parts in fine lines. 

c Grain the wood work witjra very fine pen and light indian ink, 



PL.XX. 




WOODEK STRUCTURES. 



149 



the sides of timbers as seen on a newly-planed board, the ends of 
large timbers in rings and radial cracks, and the ends of planks 
in diagonal straight lines. See also the figures at y, where the 
lines of graining outside of the knots, are to extend throughout 
the tie beam. 

d. Tint the wood work — the sides with pale clear burnt sienna, 
the ends with a darker tint of burnt sienna and indian ink. 

e. Tint the abutments, and 
other stone work, with prussian 
blue mixed with a little carmine 
and indian ink, put on in a very 
light tint. 

/. Grain the abutments in 
waving rows of fine, pale, verti- 
tical lines of uniform thickness, 
about one sixteenth of an inch 
long, leaving the upper and left- 
hand edges of the stones blank, 
to represent the mortar. The 
part of the left-hand abutment 
which is under ground is dotted 
only, as in the plate. 

g. Grain the canal walls and 
paving, as shown in the plate, 
to indicate boulder rubble. 

h. Shade the piles roughly, 
they being roughly cylindrical; 
tint them with pale burnt sien- 
na, and the shoe, S, with prus- 
sian blue, the conventional tint 
for iron. 

i. Rule the water in blue lines, 
distributed as in the figure. 

j. Tint the dirt in fine horizon- 
tal strokes of any dingy mixture, 

Note. — The above figure shows a little more than half of a queen-post 
roo/'-truss of 43 feet span. OmittiDg the light upper pieces, it may serve 
in place of Fig. 126 as a longer bridge truss; and may be drawn on any 
convenient scale from four to six feet to an inch. 




150 WOODED STRUCTURES. 

in which burnt sienna prevails, in the parts above the water, 
and ink, in the muddy parts below the water, and then add, or 
not, the pen strokes shown in the plate, to represent sand, 
gravel, &c. 

k. Place heavy lines on the right-hand and lower edges of all 
surfaces, except where such lines form dividing lines between 
two surfaces in the same plane. A heavy line on the under side 
of the floor planks, indicates that those planks project beyond 
the tie beam A. 



CHAPTER HI. 

IRON CONSTRUCTIONS. 

252. Ex. 6°. A Railway Track. PI. XXII. , Figs. 129-134. 
Mechanical construction, &c. — It may be thought an oversight 

to style this plate the drawing of a railroad track ; but taking the 
track alone, or separate from its various special supports, as bridges, 
<fcc., its graphical representation is mainly summed up in that of two 
parts ; first, the union of two rails at their joints ; second, the inter- 
section of two rails at the crossing of tracks, or at turn-outs. The 
fixture shown in Fig. 129, placed at the intersection of two rails to 
allow the unobstructed passage of car wheels, in either direction on 
either rail, is called a " Frog." Let y and z be fragments of two 
rails of the same track, then the side Hf of the point of the frog, 
and the portion k h' of its side flange, B, are in a line with the 
edges, denoted by dots, of the rails y and z, so that as the wheel 
passes either way, its flange rolls through the groove, I, without 
obstruction. When the wheel passes from y towards z there is a 
possibility of the flange's being caught in the groove, J, by dodg- 
ing the point,/". To guard against this, a guard rail, g g, is placed 
near to the inside of the other rail, supposed to be on the side of 
the frog towards Fig. 132, as shown in the small sketch, Fig. 132, 
which prevents the pair of wheels, or the car-truck, from working 
so far towards the flange, B, as to allow the flange of the wheel to 
run into the groove, J, and so run off the track. F/*, and the por- 
tion, 1 V, of the flange, A, are in a line with the inner edge of the 
rail of a turn-out, for instance, the opposite rail being on the side of 
the frog towards the upper border of the plate, as shown in Fig. 
132. Hence the flange of a car wheel in passing in either direc- 
tion on the turn-out, passes through the groove, J, and is prevented 
from running into the groove, I, by a guard rail, near the inner 
edge of the opposite turn-out rail, as at U, Fig. 132. 

253. Fig. 130 represents the under side of the right hand portion 
of the frog, and shows the nuts which secure one of the bolts which 
eecure the steel plates, as D, E ; bolts whose heads, as at u and v, 
are smooth and sunk into the plates so that their upper surfaces are 



152 IKON CONSTRUCTIONS. 

flush. It will be seen that there are two nuts on each bolt, as at 
D', on the bolt u — DD', which appears below the elevation, since it 
occurs between two of the cross-ties (sleepers) of the track. The 
nuts, as L, belonging to the bolt, b\ which are in the chairs, q'p\ 
w>', x\ are sunk in cylindrical recesses in the bottom of the frog, so 
as not to -interfere with the cross-tie on which the surface, L, rests. 
The extra nut is called a check or "jam" nut. When screwed on 
snugly it wedges the first nut and itself also against the threads of 
the screw, so that the violent tremulous motion to which the frog 
is subjected during the rapid passage of heavy trains cannot start 
either of them. 

In the end elevation, Fig. 131, A is the recess in the chair x x\ 
fitted for the reception of the rail, and B is the end of a rail in ita 
place, as shown at y in the plan. 

254. Graphical Construction. — From the above description it 
follows that the whole length of the frog depends on the shape of 
the part HyF, and the distance between this part and the side 
rails, as c I. In the present example a c— 1 ' — 11* and cf— 20". e d 
is 11" and nk is 2" from Ff. Having these relations given, and 
knowing that the lines at the extreme ends are perpendicular to the 
rails at those ends, the several figures of the frog can be constructed 
from the given measurements, without further explanation. 

255. The construction of railway-track joints so as to secure as 
nearly as possible the uniform firmness of a continuous rail, has 
long exercised the minds of railway inventors. Cast-iron chairs, 
wrought-iron chairs, long chairs resting on ties each side of the 
joint, compound rails (Div. II., 140) solid -headed, or split 
through their entire height, and fish- joints have all been used ; 
several of them in various forms. Fig. 133 is an isometrical 
drawing — scale T V — of a wooden fish-joint which allows great 
smoothness of motion and freedom from the loud clack which 
accompanies the use of ordinary chairs. A, A, A, are the sleep- 
ers (cross-ties), D is a stout oak plank, perhaps six feet long, 
resting on three sleepers, and fitted to the curved side of the rail, 
as shown at d. This plank is on the outside of the track. On 
the inner side the rails are spiked in the usual way with hook- 
headed spikes s s s, of which those at the joint, r, pass through 
a flat wrought-iron plate. P. which gives a better bearing to the 
end of the rail, and prevents dislocation of parts. Each plank, 
as D, is bolted to the rail by four horizontal half-inch bolts, 
b, b, b, b, furnished with nuts and washers on the further side of 
D (not seen). 



IR0JS" CONSTRUCTIONS. 153 

A modification of the above construction consists in substi- 
tuting for the plate P, a short piece or strap of iron fitted to the 
surface of the inside of the rail, and through which the two 
bolts bb, next to the joint, pass. 

With the now extended use of steel rails, the fish- joint, also in 
very general use, consists of an iron fish-plate on each side of the 
rail, with two bolts on each side of the joint. This makes a very 
firm joint. The plan has also been sometimes adopted of having 
the track break joints. That is, a joint, as a, Fig. 132, on one rail 
of a track, is placed opposite the centre of the rail be of the other 
line of the same track. As a track always tends to settle at the 
joints, a jumping motion is induced in a passing train, which 
perhaps may be thought to be less violent if only on one rail at a 
time. 

256. Graphical Construction. — Three lines through X, making 
angles of 60° with each other, will be the isometric axes. Remem 
bering that it is the relative position of the lines which distinguishes 
an isometrical drawing, we can place XX' parallel to the lower 
border, and thus fill out the plate to better advantage. The rail 
being 4" wide at bottom, and 4" high, circumscribe it by a square 
"Kcan, from the sides of which, or from its vertical centre line, lay 
off, on isometric lines, the distances to the various points on the rail. 
Thus, let the widest part of the rail, near the top, be 3" across, 
and -J an inch below the top ac. Let the width at the top be 2", 
and at the narrowest part 1"; and let the mean thickness of thf 
lower flange be § ". The sides of the rail are represented by tht, 
bottom lines at XX', and the tangents each side of R, to the curves 
of the section. Let the plank D be 6" wide, and 4" high. All 
the lines of the spikes, ss, are isometrical lines except their top 
edges, as st. The curve at the joint r, and at X', are similar to the 
corresponding parts of the section at X. 

To secure ease of graphical construction, let the bolt heads, b y 
&c , be placed so that their edges shall be isometric lines. 

Fig. 134, is a plan and end elevation of a heavy cast-iron 
chair designed as a partial equivalent for a continuous rail, by 
making the outside of the chair extend to the top of the rail. 
The fault in every such contrivance, the best of which at present 
seems to be the fish-joint, is that, as the joint cannot be made as 
solid as the unbroken rail, the wave of depression just in advance 
of the engine is more or less completely broken at every joint. 

258. Ex. 7°. The Hydraulic Ram. In order to give an iron 
construction, from the department of machinery, so as to render 
this volume a more fit elementary course for the machinist as well 



154 IRON CONSTRUCTIONS. 

as for the civil engineer, a simple and generally useful structure, 
viz. a hydraulic ram. has been chosen, as a fit example for the 
last to be described in detail. 

This machine is designed to employ the power of running water 
to elevate water to any desired height. 

PL XXIII., Figs. 135-13 :, shows a hydraulic ram, of highly 
approved construction, and of half the full size. 

259. Mechanical construction. — FF — FT' are feet to support 
the machine. These are screwed to a floor or other firm support. 
AB — A'B 'B ' is the inlet pipe, opening into the air chamber C, at a — a V 
and ending at del — el'cT — d"d ' the opening in the top of the waste 
valve chamber, E — E' — E". At a — a'b' is the opening as just 
noticed from the inlet pipe into the air chamber C (not seen in the 
plan). This opening is controlled by a leather valve ee\ weighted 
with a bit of copper e'e"\ and is fastened by a screw h"h'", and 
an oblong washer g'g. At X and H are the extremities of two 
outlet pipes leading from the air chamber at F"F"'. Either one, 
but not both of these outlet pipes together, may be used, as one 
of the exchangeable flanges, H' is solid, while the other is per- 
forated, as seen at M', Fig. 137. The air chamber is secured by 
bolts passing through its flange f'f\ through the pasteboard or 
leather packing, pp—p', and the flange D — D'D' at cc. This 
flange, and part of the inlet pipe are shown as broken in the 
elevation, so as to expose the valve ee', and the adjacent parts. 
LL' is a flange through which the inlet pipe passes, and this pipe 
is slit and bent over the inner edge of the aperture in LL', forming 
a flange, which presses against a leather packing, tt\ and makes a 
tight joint. The outlet pipes are secured in the same way. At 
uu — u' are the square heads of bolts which fasten the flanges to 
the projections UU — U'. K — K' is a shelf bearing the waste 
valve chamber, E — EE", and the adjacent parts. W — TV" is the 
flange of this valve chamber, secured by two bolts at vv" — v\ 
which pass through the leather packing y. h'h" is the waste 
valve, perforated with holes, se, to allow water to flow through it. 
mm' is the valve stem, d'd'k'k' is a perforated standard serving 
as a guide to the valve stem, and also as a support to the hollow 
screw s. n is a rest, secured to the valve stem by a pin p". q" 
is a nut, part of which, qq\ is made hexagonal, r is a "jam'* 
nut (253). 

In the plan of this portion of the machine, the innermost circle 
is the top of the valve stem; next is the body of the valve stem ; 
next, the top of the rest ; next, the bottom of the same; next, the 



IRON CONSTRUCTIONS. J 55 

;iut q" ; and outside of that, and resting on the top of the waste 
valve chamber, are the standards, dd. 

960. Operation. — Principles involved. — In the case of what 
might he called passive constructions, that is mere stationary sup* 
ports, like bridges, &c, a knowledge of the construction of the 
parts enables one to proceed intelligently in making a drawing ; 
but, in the case of what may, in opposition to the foregoing, be 
called active constructions, or machines, a knowledge of their mode 
of operation is usually essential to the most expeditious and accu- 
rate graphical construction of them, because a machine consists 
of a train of connected pieces, so that a given position of any piece 
implies a corresponding position for every other part. Having, 
then, in a drawing, assumed a definite position for some important 
part, the remaining parts must be located from a knowledge of the 
machine, though drawn by measurements of the dimensions of that 
part. Only fixed bearings, and centres of motion, can properly be 
located by measurement, in machine drawing. 

The principles involved in the operation of the hydraulic ram 
may be summed up under three heads, as follows: 

261. I. Work. a. When a certain weight is moved through a 
certain space, a certain amount of work is expended. 

b. Thus ; when a quantity of water descends through a certain 
space, a certain amount of work is developed. 

c, As the idea of work involves the idea both of weight moved, 
and space traversed, it follows that works may be equal, while the 
weights and spaces may be unequal. Thus the work developed by 
a certain quantity of water, while descending through a certain 
height, may be equal to that expended in raising a portion of that 
water to a greater height. 

262. II. Equilibrium, a. Where forces are balanced, or mutu- 
ally neutralized, they are said to be in equilibrium. Now the usual 
fact is, that when such equilibrium is disturbed, it does not restore 
itself at once, but gradually, by a series of alternations about the 
state of equilibrium. Thus a stationary pendulum, being swung 
from its position of equilibrium, does not, at the first returning 
vibration, stop at the lowest point, but does so only after many 
vibrations. 

b. Theoretically, these vibrations, as in the case of the pendulum, 
would never stop, but in practice the resistance of the air, friction, 
&c, make a continual supply of a greater or less amount of force 
necessary to perpetuate the alternations about the position of 
state of equilibrium. 



156 IRON CONSTRUCTIONS. 

263. HI. & physical fact taken account of in the hydraulic ram, 
is, that water in contact with compressed air will absorb a certain 
portion of such air. 

264. Passing now more particularly to a description of the ope 
ration of the hydraulic ram: 1°. Water from some elevated pond 
or reservoir flows into the machine, through the inlet pipe AA' 
and continues through the machine, and flows out through the hole 
in the waste valve h'h", pressing meanwhile against the solid parts 
of the roof of this valve, whose hollow form — open at the bottom 
— is clearly shown in Fig. 136. 

2°. Presently the water acquires such a velocity as to press so 
strongly against the roof of the waste valve, that this valve is lifted 
against the under side of the roof of its chamber which it fits 
accurately. 

3°. The water thus instantly checked, expends its acquired force 
in rushing through the valve e — e'e" and in compressing the air in 
*he air chamber C. 

4°. The holes F" or F'" of the outlet pipe, leading to an unob- 
structed, outlet, the compressed air immediately forces the water 
out through the outlet pipe until, after a number of repetitions of 
this chain of operations, the portion of the water thus expelled 
from the air chamber is raised to a considerable height. 

5°. In accordance with the second, principle, the flow of water from 
the air chamber does not cease at the moment when the confined 
air is restored to its natural density, but continues, so that — taking 
account also of the absorption of the air by the water at the time 
of compression — for a moment the air of the air chamber is more rare 
than the external atmosphere. Hence to keep a constant supply 
of air to the air chamber, a fine hole called a snifting hole, is punc- 
tured, as with a needle, at ss\ i.e., just at the entrance of the inlet 
pipe into the machine. Through this hole air enters, with a snift- 
ing sound, when the flow of water recommences, so as to supply 
the air chamber with a constant quantity of air. When the waste 
valve is at the bottom of the chamber EE', the nut and "jam" 
are together at the bottom of the screw s\ and the valve is at 
liberty to make a full stroke. By raising the valve to its highest 
point and turning the nut and "jam'' to some position as shown 
in the figure, the stroke of the valve can be shortened at pleasure, 
and, at its lowest point, will be as far from the bottom of the 
chamber as the "jam," q", is above its lowest position, 

266. In practice, it is found that the strokes of the waste valve 
shortly become regular ; their frequency depending in any given 



IKON COXSTRUCTIOXS. 157 

case on the height of the supply reservoir, the height of the ejected 
column, the size of the machine, the length of the stroke of the 
valve, &c. 

267. The proportion of water discharged into the receiving 
reservoir will also depend on the above named circumstances, 
being more or less than one third of the quantity entering the 
machine at AA'. In a machine by M. Montgolfier of France, said 
to be the original inventor — water falling 7j feet, raised g T of itself 
to a height of 50 feet. 

268. Graphical Construction. — Scale; half the full size. a. Hav- 
ing the extreme dimensions ol the plan, in round numbers 9" and 
12", proceed to arrange the ground line, leaving room for the plan 
below it. 

b. Draw a centre line, 1STC, for plan and elevation, about in the 
middle of the width of the plate. 

c. Draw a centre line, AK, for the plan, parallel to the ground line. 

d. Exactly 4^" from the centre line NC, draw the centre line vv" 
— KW for the waste valve chamber and parts adjacent. 

e. With the intersection, *, of the centre lines of the plan, as a 
centre, draw circles having radii of If" and 3 T 'g" respectively, and 
through the same centre, draw diagonals, as cc. 

f. On the centre line, NC, are the centres of the circles, F"F'", 
whose circumferences come within T T ¥ of an inch of the inner one 
of the two circles just drawn. 

g. Draw the valve, e, the copper weight e", the screw end, h, 
and the nut and oblong washer, h" and g. 

h. Locate, at once, the centres of all the small circles, cc, &c, by 
the intersection of arcs, 1" from the circle pp having * for a centre, 
with the diagonals; then proceed to draw these circles. 

i. Draw the projections, as U, drawing the opposite ones simul- 
taneously, and using an auxiliary end view of the nuts u y as often 
explained before. 

j. Draw the feet, F, with their grooves, F, and bevel edged screw 
holes, L". 

k. In drawing the shelf, K, and flange W, the intersection of the 
centre lines BK and ra'ra, is the centre for the curves which inter- 
sect the centre line AK ; the corners, 1 1, of the nuts, v,w", are the 
centres for the curves that cross the centre line, vv" ; and the 
remaining outlines of the shelf are tangents to the arcs thus drawn, 
and those of the flange are lines sketched in so as to give curves 
tangent to the arcs already drawn, and short straight lines parallel 
to vv". 



158 IKON CONSTRUCTIONS. 

I. The remaining circles and larger hexagon, u\ of this portion 
of the plan,»have the intersection of the centre lines for a centre ; and 
may be drawn by measurements independently of the elevation, or 
by projection from the elevation, after that shall have been 
finished. 

269. Passing to the elevation: — 

a. Construct, at one position of the T square, the horizontal lines 
of both feet ; then the horizontal lines of the nuts u\ and flange L', 
and projection U' ; with the horizontal lines of the floor of the air 
chamber and adjacent parts. 

b. Project up from the plan the vertical edges of the feet, FT', 
the flange, nut, and projection I/, u' and IT, the valve e\ the cop- 
per e", the screw h'\ the washer g, the air chamber flange f'f, and 
screw z. Break away the portion D — see plan — of the body of 
the machine, and the near wall of the water channel A'B'. Break 
away also the further wall of the water channel so as to show a 
section, H', of the further outlet pipe, H — see plan. Q is the centre 
of the spherical part of the air chamber to which the conical part 
is tangent. 

c. Draw all the horizontal lines of the waste valve chamber and 
parts adjacent. Make the edges of the threads of the screw straight 
and slightly inclined upwards toward the right. 

d. Project up from the plan, or lay off, by measurement, the 
widths of various parts through which the valve stem passes, and 
draw their vertical edges. 

Fig. 136 is a section of the waste valve chamber, showing part 
both of the interior and exterior of the waste valve. The dotted 
circles form an auxiliary plau of this valve, in which the holes have 
two radial sides, and two circular sides with x" as a centre. The 
top of the valve is conical, so that in the detail below, two of the 
sides of the hole n, tend towards the vertex, x. At n\ one of these 
holes, of which there are supposed to be five, is shown in section. 

Fig. 137. The outlines of M, one of the outlet pipe flanges, are 
drawn by processes similar to those employed in drawing the shelf, 
K, in plan. 

270. Execution, As a line drawing, the plate explains itself. It 
would make a very beautiful shaded drawing and one that the 
careful student of the chapter on shading and shadows, would be 
able to execute with substantial accuracy, without further instruo 
tion. 

We conclude this division with the following additional exer- 
cises as examples of iron constructions — one from civil engineer- 



IRON CONSTRUCTIONS. 



159 



ing practice, the other three from mechanical engineering; styl- 
ing them exercises, since, being partially shown (yet, with the 
description, sufficiently so for their purpose), they leave some- 
thing to be supplied by the student from the general insight 
gained from previous practice. 

Exercise 1. A Stop-valve. — The following figure shows one of many- 
forms of valve differing more or less in detail, and made for the purpose 
of shutting off the passage of steam, water, etc., through pipes. Such 






K. 




valves either lift from their seats, as in the example shown, or slide off 
them, in which case they are sometimes called gates. 

The figure represents what is called a globe-valve, from the general 
external form of its valve-chamber NCCL. In this chamber is a bent par- 
tition, or diaphram, CEC, containing the seat, E, of the valve D. This 
valve is raised or lowered by means of the hand-wheel K, and screw 
valve-stem F working in the collar G, which is screwed into the top, 
NC, of the chamber. 

The head at the bottom of the valve-stem, working loosely in the hol- 
low head of the valve, raises the latter vertically without turning it. 
The cap H secures the necessary packing. Opening the valve then 
allows of the passage of any fluid through AB and the pipes which may 



160 IRON C OBSTRUCTIONS. 

be attached at A and B. These openings are from \" to 2" diameter. 
The measurements and scale may therefore be assumed, and plan and 
end elevation added. 

Exercise 2. An iron truss bridge. Pi XXIV., Figs. 1-7. — This bridge 
is partly of wrought, and partly of cast iron, and known from its form 
and its inventor as Whipple's trapezoidal-truss bridge. 

The upper chords, a — a'a', are hollow cast-iron cylinders 7£" diameter, 
and |" to %" thickness of metal. The posts, p', and struts, 3, S'S", are 
also of cast-iron, the latter, double, as seen in the fragment of end ele- 
vation, Fig. 2, and fragment of plan, Fig. 3. 

The posts extend through the flooring, where they are 5" in diameter, 
and rest on seats on the tops of the cast-iron coupling blocks, n'n' , as 
shown in the plan, Fig. 5, and end elevation, Fig. 6, of one of these 
blocks. 

The lower chord, bb — b'V, is composed of heavy wrought-iron rods made 
in links embracing two successive coupling-blocks, in the manner shown 
in Figs. 4-6. The two end lengths, however, are single, as shown, 
and are secured by nuts, g ', at the outer end of the shoes s, s", which 
holds the feet of the struts SS'S". 

The structure is further held in shape, and the forces acting in it suit- 
ably sustained and distributed by the diagonal and vertical rods rV, 
each of which, after the first two from the end, crosses two panels of the 
bridge, as the spaces between the posts are called. 

The horizontal diagonal rods, f, under the floor, tightened by links I 
working on right- and left-handed screws (Div. V., Ex. 9) in the adja- 
cent rod ends, provide against the horizontal force of winds. The light 
transverse flanged beams h — k", overhead, also help to stiffen the struc- 
ture laterally. 

The main transverse beams c — c' — c" rest on the coupling-blocks, and 
support the floor joists dd'd", on which the floor planks gg'g" rest. 
CCC" is the coping, serving to cover the irregular ends of the floor 
planks, and as a guard to prevent vehicles from striking the truss. 

Fig. 7 is an enlarged view of the centre joint where the two halves of 
the posts meet. 

Other useful details would be vertical longitudinal sections of the 
joints as ee" and e' , which would show an opening in the under side of 
the upper chord, sufficient for the entrance of the diagonal rods, and 
these rods forged into rings clasping the stout wrought-iron pins e, e'e"; 
also the level bearing for the head of the post, except at the joint at the 
head of the strut. 

The three top cross-beams indicated at kh, Fig. 3, show that a skew- 
bridge is represented, that is, one which crosses the stream obliquely, 
the extreme timber, h, being parallel to the length of the stream. 



IRON" CONSTRUCTIONS. 161 

The span of the bridge is 114 leet, in 12 equal panels of 9|- feet each; 
the roadway is 19 feet wide from centre to centre of the trusses, which 
are 15 feet 9 inches in height from the centre of the coupling-blocks, n\ 
to that of the upper-cl>ord pins as at e'. 

Suitable scales are 3 to 5 feet to 1 inch for the general views, and 
from G inches to 1 foot to one inch for the details. 

Exercise 3. A vertical ooiler. PL XXIV., Fig. 8. — This figure, being 
given partly as an excellent example in shading, and of certain flame 
effects instructive to the draftsman, is described without letters of ref- 
erence. 

The figure represents a vertical section of what is known as the Shap- 
ley patent boiler, differing from the ordinary tubular vertical boiler as 
appears from the figure and following description. 

The central combustion chamber, being tall, is designed to effect 
three results ; viz., to raise its top, called the crown sheet, so far above 
the fire as to retard burning out ; to afford abundant room for perfect 
combustion, thereby generating more heat; and to effectually convey 
this heat to the water which surrounds the fire-box in a thin sheet. 

Heat is further conveyed to the water by passing, as shown by the 
arrows, through short transverse tubes, two of them seen in section, and 
vertical tubes between the fire-box and the outer shell. These open into 
the annular base flue (interrupted by the ash-pit door), which leads to 
the smoke-pipe (sometimes called the up-take). 

The upper section, or steam-dome, is mostly occupied by steam, and 
is stayed by bolts to the crown sheet. 

Since the tubes, when sooty, lose much of their heat-conducting power, 
they are, in this boiler, made very easily accessible for frequent clean- 
ing by connecting the two sections of the boiler by a double annular 
jacket which contains no steam or water and sustains no pressure. . It is 
made in sections for easy removal, and thus allows ready access to the 
tubes. 

Exercise 4. A direct- acting steam-pump. PI. XXIV., Fig. 9. — The mag- 
nitude and variety of pumping requirements for water, oil, and various 
other liquids, hot or cold, thin or viscid, pure or gritty, and for drain- 
age, mining, city, hotel, railroad - station, and other purposes, have 
called forth a large amount of inventive talent and many ingenious and 
effective pumping engines. 

The .figure represents a vertical longitudinal section of the Knowles 
steam-pump, affording a useful study and guide in making a finished 
drawing. 

BB is the pump barrel or water cylinder — lined, when the character 
of the fluid to be pumped requires it, with composition linings, shown 
at xx and similarly in section on the upper side of the barrel. P is the 



162 IRON CONSTRUCTIONS. 

water piston with its packing p, and secured to the piston-rod a by 
nut and lock-nut (Div. V., 12) seen at the left. 

IEF is the steam cylinder with its piston on the same piston-rod, a, 
with the water piston, thus forming what is called a direct-acting pump. 
Both cylinders are provided with stuffing-boxes ~hg and K. 

The pump valves under the letter o are here shown as lifting disk 
valves circular in plan, but may be cage, or hinge, or any other valves. 

In the position shown, and the piston still moving towards the left, 
water is entering through the lower or suction circular inlet and the 
lower right-hand valve, and is discharging by the upper or discharge 
pipe, which is smaller than the suction pipe. By raising the upper left- 
hand valve the discharge water also partly enters the air-chamber A, 
where, by compressing the confined air, a steady discharge is obtained. 
The valves rise and fall, each working on a short spindle, and are 
quickly closed by the aid of spiral springs above them ; seen on the two 
closed valves. 

The steam and exhaust ports and passages to the steam cylinder are 
of the usual form ; n, the orifice for the admission of steam from the 
boiler, and the central orifice is the exhaust. The steam-valve is a 
double D valve. 

A stroke to the right being about to begin, a roller on the opposite 
side of the tappet arm CC, carried by the piston-rod a, raises the left 
end of the rocker DD. This, by means of the link s, slightly rotates the 
valve-rod I and its "chest-piston," Fig. 11, so as to bring it into a posi- 
tion to take steam through the small passage at the lower right-hand 
corner of the steam-chest Gr, which throws the piston to the opposite end 
of its stroke, carrying the valve by means of its stem T, Fig. 10. Steam 
can then enter the left-hand end of the cylinder through the left-hand 
chamber of the D valve, while exhaust steam escapes through the pas- 
sage y and the right-hand chamber into the central, or exhaust passage. 

At i is the valve-rod guide, jis a collar on the valve-rod. u clamps 
the rocker connection to the valve-rod. t adjusts the link s. M is the 
oil-cup, and n a stud to attach a hand lever. 

These pumps are made of a large range of sizes, from water cylinders 
of 2", and steam cylinders of 3V' diameter, and 4" stroke; to water 
cylinders of 20", and steam cylinders of 28" diameter, and 12" stroke. 
Fig. 9 may be regarded for drawing purposes as a sketch (from a scale 
drawing and in true proportion, however) of a pump having a water 
cylinder of 7", and a steam cylinder of 12" diameter, with a 12" stroke. 

For variety of practice in the use of scales, the pump may then be 
drawn on a scale of i or -^ the full size, with details on scales of 3" to a 
foot, or of full size. 

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INDUSTRIAL SCIENCE DRAWING. 

"In all the constructive trades, the greater part of a workman's instruction* 
■are given to him in the form of working drawing*. Yet ice suffer the budding 
artisan to pass througli the schools ignorant of the first rudiments of a science 
that is as essential to his work as are the four rules of arithmetic" — Prof. S. P. 
Thompson, in Pop. Science Monthly for Xov. 1880. 

"The practice being to lay before each student a drawing of some part or parts of a 
structure, which he is requested to copy .... and the same course is pursued 
until he . . . . eventually is enabled to make a very creditable, or even a highly finished 
drawing from copy. If, "however, at the end of one or two years practice he is asked to 
make side and end elevations and sections of his lead pencil, or instrument box, the 
chances are that he can neither do the one nor the other. Strange as it may appear, 
this is a state of things winch I have had frequent opportunities of witnessing in the 
case of students who have excellent specimens of their own drawings from copy .... 
with all the shadows admirably projected, being, however, perfectly ignorant of the 
rides for projecting such shadows. ' "—"William Bums, Assoc. Inst. C. E., Lo>T>ojr. 






New York, January, 1881. 
In connection with the above extracts, and similar ones which might be 
quoted, on the wide importance of elementary instruction in mechanical 
drawing, and the equal importance of a right method in teaching it, 

JOHN WILEY <fe SONS, Publishers, 

15 Astor Place, New York, 

respectfully invite the attention of School Officials, Teachers, Stu- 
dents and Artisans, to the new and enlarged edition, now ready, of the 

ELEMENTARY PROJECTION DRAWING, 

AXD TO THE OTHER ELEMENTARY WORKS ON 

INDUSTRIAL SCIENCE DRAWING, 

By S. EDWARD TVARREX, C.E., 

Former Professor in the Rensselaer Polytechnic Institute, and elsewhere. 



DESCRIPTIONS. 

ELEMENTARY PROJECTION DEAWHSTG-, 

Fifth Edition, Revised and Enlarged, with preliminary instruc- 
tions on drafting instruments, and a new division on 
Elements of Machines. 

Div. I. Projections, of simple solids, prisms, pyramids, cylinders, 
cones and spheres, and their intersections and developments. 4 plates. 

Div. II. Wood, Mason r y and Metal Details, Carpentry 
Joints, etc., to be drawn to scale from measurements. -I plates. 

Div. III. Elementary Shadows and Shadiny, sufficient for 
ordinary practice, and with new examples. 3 plates. 

Div. IV. Isometrical and Oblique Projections, or Mechanical 
Perspective, easily learned and universally useful. 4 plates. 

Div. V. (New) Elements of Machines. Cranks. Eccentrics, Toothed 
Wheels, Screws, etc. 4 plates. 

Div. VI. Elementary Structures and Machines. An appro- 
priate, practical summary of the preceding divisions, and with valuable 
new examples, a Steam Pump, an Iron Bridge, etc. 5 plate-. 

^P" This volume, and the one on Drafting Instruments, described below, are 
especially commended to Preparatory Scientific Schools, Hiirh Schools and Academies. 
and Self -Instructors, as generally first in importance when more cannot be attempted. 
Both contain the valuable feature of suitable examples for practice, as usual in arith- 
metics, etc., to test the pupil's understanding of the subject. 

12mo. Cloth. 24 Dlates. fcl.50. 



OTHER ELEMENTARY VOLUMES. 



FREE-HAND GEOMETRI- 
CAL DRAWING. (Revised 
and enlarged edition.) Part I. 
Plane Drawing. Part II. Solid 
Drawing. Part III. Elements of 
Geometric Beauty. A suitable 
special training of the mechanical 
draughtsman and letterer, in free- 
hand drawing of regular forms, 
structure and machine sketching 
and lettering. 12 plates and many 

cuts $1 00 

Also in four 4to numbers, each 
30 cents. 



ELEMENTARY LINEAR 
PERSPECTIVE, both of 
Forms and Shadows; or, the Re- 
presentation of objects as they ap- 
pear, made from the Representa- 
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two Parts — I. Primitive "Methods, 
with an Introduction. II. Deriv- 
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Aerial Perspective. Wood engra- 
vings. 12mo, cloth $1 00 

This volume, complete in itself, and 
with full explanations of the principles 
and rules of Perspective, is useful to all 
who, whether for husiness or pleasure. 
have occasion to do with the arts of orna- 
mental or pictorial design— Artists, Archi- 
tects, Decorators, Engravers, and in La- 
dies' Seminaries and Schools of Design. 



DRAFTING INSTRUMENTS 
AND OPERATIONS. (Re- 
vised edition, enlarged, and with 
a selection of the most commonly 
useful plane problems, and beau- 
tiful new plates.) In four Divi- 
sions — I. Instruments and Mate- 
rials. II. Fundamental Opera- 
tions. III. Plane Problems and 
Practical Exercises. IV. Ele- 
ments of Taste in Geometrical 
Drawing. 7 plates and additional 
woodcuts 

This volume teaches correct practice in 
the execution of finished mechanical 
drawings, by full instructions, applied in 
many varied exercises, which always in- 
terest the student by their pleasing effect 
and obvious utility. 

1 vol. 12mo, cloth $1 25 

Or in four Parts, 4to, neat 

cover 1 25 

(Parts may be had separately.) 

N. B— These Works, together icith cheap suitable " School Sets" of Drawing Instal- 
ments, can be obtained through booksellers everywhere. 

These works are believed from long experience, and the testimony of many corres- 
pondents during the past twenty years, to be at once elementary in matter, and tho- 
rough but simple in treatment ; perfectly adapted to preparatory scientific instruction, 
and yet, so far as they go. giving reasons and principles along with abundant practice. 
rather than plates for imitation with arbitrary directions only ; the author having dis- 
carded the copying system as comparatively worthless, after short trial, more than 
twenty -five years ago, and substituted therefor regular construction exercises conduct- 
ed by means of explanatory lectures, subsequently wrought up into text -books. 

^P*" Each work, moreover, is complete in itself, and thus adapted for separate use, 
and especially for self -instruction. 



PLANE PROBLEMS IN EL- 
EMENTARY G-EOMETRY; 

or, Problems on the Elementary 
Conic Sections — the Point, 
Straight Line, and Circle. In two 
Divisions — I. Preliminary or In- 
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metrical Problems. "With an In- 
troduction, plates, and woodcuts. 

12mo $1 25 

A useful companion to the study of Geo- 
metry, to cultivate observation, neatness, 
accuracy, and memory of the principles. 
This volume, complete in itself, contains 
a full and varied separate collection of 
problems, especiallj- on the interesting 
and practically useful subject of tangen- 



HIGHER WORKS. 



By the same author, for Colleges and Technical Schools, Engineers, Masons, and 
Machinists. 



THE ELEMENTS OF DE- 
SCRIPTIVE GEOMETRY, 
SHADOWS AND PER- 
SPECTIVE. With a brief 
treatment of Trihedrals, Transver- 
sals, and Spherical, Axonometric 
and Oblique Projections. For Col- 
leges and Scientific Schools. 1 vol. 
small 8vo, with 24 folding plates 
in separate case $3 50 



II. 
ELEMENTS OF DESCRIP- 
TIVE GEOMETRY. Com- 
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with the subsequent volumes of 
application, for Technical Schools. 
This differs from L, in being a 
fuller treatise on Descriptive Geo- 
metry only, is limited to surfaces 
of revolution, and yet fully ex- 
hibits the principles and opera- 
tions of the subject. One vol. 
8vo, 24 folding plates and wood- 
cuts, cloth $3 50 



in. 
GENERAL PROBLEMS OF 
SHADES AND SHADOWS. 

Formed both by Parallel and by 
Radial Rays; and shown both in 
Common and in Isometrical Pro- 
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of Shading. 8vo, with 15 folding 
plates, cloth $3 00 



IV. 

HIGHER PERSPECTIVE, 
GENERAL PROBLEMS IN 



THE LINEAR PERSPEC- 
TIVE OF FORM, SHAD- 
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or, the Scenographic Projections 
of Descriptive Geometry. 8vo, 
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v. 
ELEMENTS OF MACHINE 
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DRAWING ; or, Machine 
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and for the use of Mechanical Es- 
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practice. In two vols. 8vo, 1 vol. 
of text illustrated with 119 fine 
wood engravings, and 1 vol. con- 
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cloth $7 50 



vX 
STEREOTOMY— PROB- 
LEMS IN STONE-CUT- 
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JOHN WILEY & SONS 

PUBLISH THE 

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This work consists chiefly of the Lectures delivered by Professor Wood upon 
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A TREATISE ON THE THEORY OF THE CONSTRUCTION 
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